The US Democratic Party is set to hold the last two primary elections in its marathon presidential nomination race.
As Montana and South Dakota go to the polls, Barack Obama is widely thought to be poised to become the first black nominee of a major US political party.
Speculation is rife that rival Hillary Clinton may be close to pulling out.
Mr Obama, who has won more pledged delegates and has the party nomination within reach, is already eyeing the general election in November.
He is just more than 40 delegates short of the 2,218 needed to clinch the Democratic nomination at the August convention where the candidate is selected.
While there are only 31 elected delegates at stake on Tuesday, Mr Obama could gain the support of enough of the remaining uncommitted "super-delegates" - party officials with a free choice over who to support - to pass the winning post.
The real drama will start in the moments after Tuesday's polls close, says BBC North America Editor Justin Webb.
That is when Mr Obama is hoping for a flood of endorsements from so far undecided super-delegates, our correspondent says, and such a flood could, within hours, force Mrs Clinton to end her bid.
Mr Obama picked up several endorsements from super-delegates on Monday, and the backing of South Carolina Representative Jim Clyburn, an influential African-American politician, was expected to follow on Tuesday.
Mrs Clinton's staff have indicated that she will stay in the race for the time being, but the overwhelming expectation is that she will pull out by the end of the week.
There are reports that members of her team have been asked to hand in their expenses.
She is expected to spend election night in New York, her home state.
'Extraordinary race'
Mr Obama is due to speak in St Paul, Minnesota, where the Republicans will hold their convention in September and formally nominate Senator John McCain as their candidate.
This can be read as a clear signal that he has his mind now on the task ahead - the general election in November, our correspondent adds.
Mr Obama appears well aware that he must quickly bring Mrs Clinton's disappointed supporters on board and heal divisions within the party.
He told reporters in Michigan that in a call to Mrs Clinton following her weekend win in Puerto Rico's primary, he had congratulated her on running "an extraordinary race" and working incredibly hard.
"I told her that once the dust settled I'm looking forward to meeting with her at a time and place of her choosing," he said.
"The sooner we can bring the party together, the better, so we can focus on John McCain and taking back the White House."
But efforts to bring the Democratic contest to a gentle end have been hampered by a new outburst from former President Bill Clinton.
He accused Mr Obama of getting other people to smear Mrs Clinton, and the press of being pro-Obama and providing the most rigged election coverage in modern history.
(BBC)
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