However The Associated Press news agency says Obama already has enough combined pledged delegates and superdelegates to clinch the nomination.
The AP's tally includes delegates whose endorsements have not yet been publicly declared.
As voters went to the polls, AP also reported that Clinton, the New York senator, was to acknowledge after the primaries that Obama had won enough delegates for the nomination.
Clinton's campaign on Tuesday swiftly denied the reports, with Terry McAuliffe, Clinton's campaign manager, telling CNN the reports were "absolutely not" true.
AP later reported that Clinton had said in a conference call with other New York politicians that she was open to a vice-presidential nomination with Obama, one of the call's participants told the news agency, if it was felt it would help the Democrats in the November presidential election.
But Maria Cardona, Clinton's national campaign spokeswoman, told Al Jazeera both claims were untrue, and said the New York senator felt she owed it to her supporters "to fight until the very end".
"She is not going to concede tonight ... Obama does not have the numbers and until someone has that magic number she is going to continue to press her case," she said.
The news came as Jimmy Carter, the former US president and a senior Democrat "superdelegate," said on Tuesday that he would endorse Obama, a huge boost for the Illinois senator.
A total of 12 superdelegates have endorsed Obama on Tuesday.
Meanwhile, a total of 31 delegates are at stake in Montana and South Dakota.
Results are expected from Montana sometime after 9pm local time (02:00 GMT) when polls close and from South Dakota, which covers two time zones, from after 7pm (00:00 GMT) and 8pm (01:00 GMT).
About 180 superdelegates - senior party officials who can choose to back either candidate at August's convention in the state of Denver - are still to declare their support.
However, many analysts feel that their endorsements could start to stream in once the state nominating contests are over, possibly clinching the contest for Obama.
On Tuesday, Obama also gained the endorsement of US representative James Clyburn, the highest ranking African-American in the US Congress.
Rob Reynolds, Al Jazeera's Washington correspondent, said earlier on Tuesday that it does appear that, bit by bit, Obama is closing in on the Democrat nomination and by Tuesday evening, if he has enough superdelegates, he could well stand before a crowd in Montana and say he has won the nomination.
Clinton is due to head back to her senate seat of New York for an event on Tuesday evening, prompting speculation that she was planning to end her campaign.
Bill Clinton, her husband and former US president, also sounded as though the campaign was winding down when he spoke to voters in South Dakota.
"This may be the last day I'm ever involved in a campaign of this kind," he said. But the former first lady said that Tuesday marked "the beginning of a new phase of the campaign".