Mr Obama will speak in St Paul, Minnesota once the polls are closed
US Democratic presidential hopeful Barack Obama will clinch his party's nomination once the final primaries are complete, US media report.
Voters in South Dakota and Montana have been voting in the party's final nominating contests.
Based on exit poll data and information from US media, Mr Obama will secure enough delegates to win the candidacy.
His rival Hillary Clinton has said she will not concede tonight, but is "open" to becoming Mr Obama's running mate.
Mrs Clinton made the comments in a conference call with other New York lawmakers, US media said.
Montana and South Dakota are completing the primaries, with Mr Obama only a few delegates short of the 2,118 needed ahead of the polls closing.
If he does as well as expected in Tuesday's contests - where 31 elected delegates are at stake - he will need to pick up only a handful of endorsements from the remaining uncommitted "super-delegates" - party officials with a free choice over who to support - to pass the winning post.
Polls were due to close at 1900 (0000 and 0100 GMT) across two time zones in South Dakota and at 2000 (0200 GMT) in Montana.
Former President Jimmy Carter has confirmed he is endorsing Mr Obama.
"The Obama people already know they have my vote when the polls close tonight," he told the Associated Press (AP) news agency.
In the conference call, Democratic Party Representative Nydia Velazquez said Mr Obama would win key voting blocs such as Hispanics if Mrs Clinton were on the ticket as vice-president.
Mrs Clinton replied: "I am open to it."
Earlier, AP reported that two senior campaign officials were saying the race was effectively over.
The officials suggested that in a speech in New York City Mrs Clinton would not formally end her campaign but would accept Mr Obama had enough delegates to win, the agency said.
Not formally conceding would give Mrs Clinton more leverage on her future role in the Democratic Party's battle against Senator John McCain in the November election, analysts said.
But, speaking on CNN, Clinton campaign chief Terry McAuliffe said the reports were "100%" incorrect.
"I can honestly say as chairman of this campaign that until someone has the numbers, this campaign carries on.
"The race goes on. We've got two important votes today and then tomorrow we've got to work the super-delegates and we believe we can persuade enough super-delegates to come over and support Senator Clinton."
Drama still lies ahead and will start in the moments after Tuesday's polls close, says BBC North America editor Justin Webb.
Bill Clinton addressed a crowd in Milbank on Monday
Mr Obama picked up several other endorsements from super-delegates on Monday, and the formal backing of South Carolina Representative Jim Clyburn, an influential African-American politician, on Tuesday.
Campaigning in South Dakota on Monday, Mrs Clinton said the end of voting would mark "the beginning of a new phase of the campaign" - one in which she would turn her attention to winning over super-delegates.
Mr Obama is due to speak in St Paul, Minnesota, where the Republicans will hold their convention in September and formally nominate Mr McCain as their candidate.
Mr Obama appears well aware he must heal divisions within the Democratic Party.
"The sooner we can bring the party together, the better, so we can focus on John McCain and taking back the White House," he said.
Are you in South Dakota or Montana? Are you voting on Tuesday? How do you intend to vote? What outcome do you expect to see after the final round of primaries? Send us your comments using the form below.
The BBC may edit your comments and cannot guarantee that all emails will be published.
This page is best viewed in an up-to-date web browser with style sheets (CSS) enabled. While you will be able to view the content of this page in your current browser, you will not be able to get the full visual experience. Please consider upgrading your browser software or enabling style sheets (CSS) if you are able to do so.