Scott Brown is already a state senator in Massachusetts
US President Barack Obama has been campaigning in Massachusetts ahead of the by-election to fill the late Ted Kennedy's Senate seat.
Opinion polls suggest that Martha Coakley, the candidate from Mr Obama's Democratic Party, is locked in a dead heat with her Republican rival.
Scott Brown is hoping that on Tuesday he will win the Republicans' first Senate seat in the state since 1972.
Defeat for the Democrats could mean the blocking of the Obama healthcare bill.
Mr Brown has promised to vote against it and his victory in the by-election - or special election, as it is known in America - could deprive the Democrats of the 60-vote majority they need to pass that and other legislation.
The Obama administration has made healthcare reform its top domestic priority.
The unexpectedly tight Senate race in a state where registered Democrats outnumber Republicans three to one reflects a nasty, anti-establishment environment ahead of the November Congressional elections in which Democrats hope to hang onto their majority, the Associated Press reports.
The significance of the vote is all the greater as it falls a day before the first anniversary of Mr Obama's inauguration.
Tapping into anger
"Understand what's at stake here, Massachusetts," Mr Obama told a Coakley election rally in Boston on Sunday.
Martha Coakley is the state attorney general for Massachusetts
"It's whether we're going forward or going backwards. If you were fired up in the last election, I need you more fired up in this election."
Mr Brown has been able to tap into voter anger and anxiety over federal spending, expanded government and high unemployment, as well as healthcare reform.
"It's us against the machine," he told supporters on Sunday in Worcester, the state's second-largest city.
"The establishment is afraid of losing their Senate seat. You can all remind them that this is not their seat, it is yours."
Since Kennedy's death last year, his seat has been held by an interim appointee, Senator Paul G Kirk Jr.
Massachusetts officials say it could take more than two weeks to certify the election results, which may give Democrats enough time to push the healthcare legislation through Congress, AP adds.