28 February 2012 Last updated at 18:51 GMT
Both men have been campaigning intensively over the past few days. Latest polls give Mr Romney a marginal lead in Michigan, and a stronger advantage in Arizona.
Newt Gingrich and Ron Paul are focusing their efforts on other states.
Analysts say a victory in his home state of Michigan is key for Mr Romney.
He has long been seen as the front-runner and favourite for the nomination - and currently leads the race for delegates - but has struggled to win over a strong majority of conservative Republican voters.
The winner of the eventual nomination will go on to face President Barack Obama in the November election.
Momentum
On Tuesday, Mr Romney appeared to acknowledge that he has had trouble winning over conservative voters in a state where he was expected to do well.
Mr Romney said his disconnect with the party's right-wing stemmed from his unwillingness to make "incendiary" comments.
He accused his rivals of saying "outrageous things" in an effort to win the backing of the Republican base, adding that he was not prepared to set his "hair on fire" in a bid for support.
He also attacked Mr Santorum's recent move to target Democratic voters with an automated message criticising Mr Romney's record on bailouts for the automobile industry.
Mr Romney accused Mr Santorum of trying to "kidnap the primary process" by attempting to turn Democrats against him. He told Fox News on Tuesday morning that the tactic was "outrageous and disgusting... a terrible, dirty trick".
"This is a new low for his campaign and that's saying something," Mr Romney said.
Mr Santorum's recorded message to Democrats said: "Romney supported the bailouts for his Wall Street billionaire buddies but opposed the auto bailouts.
"That was a slap in the face to every Michigan worker and we're not going to let Romney get away with it," it continued.
Although only Republicans may participate in Michigan's primary, electoral rules allow voters to temporarily change their affiliation on the spot - giving Democrats the opportunity to cast a ballot on Tuesday.
Mr Santorum and his supporters have also spent about $2m (£1.3m) on advertising in Michigan.
On the campaign trail Mr Romney had played up his ties to Michigan, where his father was a former governor.
Whoever wins in Michigan could gain crucial momentum ahead of next week's "Super Tuesday" votes, which sees 10 states go to the polls.
Mr Gingrich, a former Speaker of the House of Representatives, and Texas Congressman Ron Paul are focusing their efforts on next week's vote.
Focus on economy
After spending much of the past week campaigning on social issues, on Monday Mr Romney, a former Massachusetts governor, turned his focus on the economy.
He challenged former Pennsylvania Senator Rick Santorum for putting social issues at the centre of his campaign, saying: "If the economy is going to be the issue we focus on, who has the experience to actually get this economy going again?"
Mr Romney, who used to run a successful private equity firm, told supporters at a campaign event that Mr Santorum was a nice guy but he would not be able to create jobs.
Meanwhile, on the campaign trail, Mr Santorum told supporters it was a "joke" for Mr Romney to attack him for not being a real conservative.
The former senator also derided "climate science" and Wall Street bailouts - referring to legislative positions the former governor had supported.
He used an editorial in the Wall Street Journal to highlight his own principal economic initiatives.
Continue reading the main story
His editorial followed remarks slamming Mr Romney at the weekend, who Mr Santorum described as "uniquely unqualified" to take on the key issues facing America.
Precarious lead
In recent weeks, Mr Santorum has mounted a strong challenge to Mr Romney in Michigan.
An average of polls in the state compiled by Real Clear Politics shows the former governor clinging to a narrow lead of 1.5%, although he maintains a more comfortable lead in Arizona.
A loss for Mr Romney in either state could establish Mr Santorum as a new frontrunner in the presidential race, correspondents say, and raise questions about Mr Romney's ability to appeal to his party's base.
Mr Santorum sprang an upset in the last round of voting, when he picked up three victories - in Missouri, Minnesota and Colorado - in a single night.
Mr Romney currently has 123 delegates, compared to Mr Santorum's 72, with 1,144 needed to secure the nomination.
Fifty-nine delegates are at stake in Tuesday's primary elections: 29 from Arizona and 30 from Michigan.COPY : http://www.bbc.co.uk/
Mitt Romney and Rick Santorum are going head-to-head as US voters in Michigan and Arizona choose their picks for Republican presidential candidate.
Both men have been campaigning intensively over the past few days. Latest polls give Mr Romney a marginal lead in Michigan, and a stronger advantage in Arizona.
Newt Gingrich and Ron Paul are focusing their efforts on other states.
Analysts say a victory in his home state of Michigan is key for Mr Romney.
He has long been seen as the front-runner and favourite for the nomination - and currently leads the race for delegates - but has struggled to win over a strong majority of conservative Republican voters.
The winner of the eventual nomination will go on to face President Barack Obama in the November election.
Momentum
On Tuesday, Mr Romney appeared to acknowledge that he has had trouble winning over conservative voters in a state where he was expected to do well.
Mr Romney said his disconnect with the party's right-wing stemmed from his unwillingness to make "incendiary" comments.
He accused his rivals of saying "outrageous things" in an effort to win the backing of the Republican base, adding that he was not prepared to set his "hair on fire" in a bid for support.
He also attacked Mr Santorum's recent move to target Democratic voters with an automated message criticising Mr Romney's record on bailouts for the automobile industry.
Mr Romney accused Mr Santorum of trying to "kidnap the primary process" by attempting to turn Democrats against him. He told Fox News on Tuesday morning that the tactic was "outrageous and disgusting... a terrible, dirty trick".
"This is a new low for his campaign and that's saying something," Mr Romney said.
Mr Santorum's recorded message to Democrats said: "Romney supported the bailouts for his Wall Street billionaire buddies but opposed the auto bailouts.
"That was a slap in the face to every Michigan worker and we're not going to let Romney get away with it," it continued.
Although only Republicans may participate in Michigan's primary, electoral rules allow voters to temporarily change their affiliation on the spot - giving Democrats the opportunity to cast a ballot on Tuesday.
Mr Santorum and his supporters have also spent about $2m (£1.3m) on advertising in Michigan.
On the campaign trail Mr Romney had played up his ties to Michigan, where his father was a former governor.
Whoever wins in Michigan could gain crucial momentum ahead of next week's "Super Tuesday" votes, which sees 10 states go to the polls.
Mr Gingrich, a former Speaker of the House of Representatives, and Texas Congressman Ron Paul are focusing their efforts on next week's vote.
Focus on economy
After spending much of the past week campaigning on social issues, on Monday Mr Romney, a former Massachusetts governor, turned his focus on the economy.
He challenged former Pennsylvania Senator Rick Santorum for putting social issues at the centre of his campaign, saying: "If the economy is going to be the issue we focus on, who has the experience to actually get this economy going again?"
Mr Romney, who used to run a successful private equity firm, told supporters at a campaign event that Mr Santorum was a nice guy but he would not be able to create jobs.
Meanwhile, on the campaign trail, Mr Santorum told supporters it was a "joke" for Mr Romney to attack him for not being a real conservative.
The former senator also derided "climate science" and Wall Street bailouts - referring to legislative positions the former governor had supported.
He used an editorial in the Wall Street Journal to highlight his own principal economic initiatives.
Continue reading the main story
US politics glossary
Use the dropdown for easy-to-understand explanations of political terms:
Primary
A state-level election held to nominate a party's candidate for office. Regulations governing them and the dates on which they are held vary from state to state. In some states, voters are restricted to choosing candidates only from the party for which they have registered support, however 29 states permit open primaries in which a voter may opt to back a candidate regardless of their nominal affiliation. In this case, strategic voting may take place with, for example, Republicans crossing over to back the perceived weaker Democratic candidate. Primaries first emerged as a result of the so-called progressive movement of the early 20th Century, which argued that leaving the nomination process purely to party bosses was inherently undemocratic.
In it, he took aim at his rival, saying Mr Romney was "attempting to distract from his record of tax and fee increases as governor of Massachusetts, poor job creation, and aggressive pursuit of earmarks", and that Mr Romney's plans did not go far enough.
His editorial followed remarks slamming Mr Romney at the weekend, who Mr Santorum described as "uniquely unqualified" to take on the key issues facing America.
Precarious lead
In recent weeks, Mr Santorum has mounted a strong challenge to Mr Romney in Michigan.
An average of polls in the state compiled by Real Clear Politics shows the former governor clinging to a narrow lead of 1.5%, although he maintains a more comfortable lead in Arizona.
A loss for Mr Romney in either state could establish Mr Santorum as a new frontrunner in the presidential race, correspondents say, and raise questions about Mr Romney's ability to appeal to his party's base.
Mr Santorum sprang an upset in the last round of voting, when he picked up three victories - in Missouri, Minnesota and Colorado - in a single night.
Mr Romney currently has 123 delegates, compared to Mr Santorum's 72, with 1,144 needed to secure the nomination.
Fifty-nine delegates are at stake in Tuesday's primary elections: 29 from Arizona and 30 from Michigan.COPY : http://www.bbc.co.uk/
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