Clinton's campaign on Tuesday denied news reports that she would concede following the two final primaries, after the Associated Press news agency quoted her campaign officials as saying she would acknowledge after the final votes that Obama has clinched enough delegates to win the nomination.
Terry McAuliffe, Clinton's campaign manager, told CNN the reports were "absolutely not" true and that the earlier AP reports she would concede Obama had the delegate numbers were "100 per cent incorrect".
"No one has the number to be the nominee of the Democratic party right now," he said.
A total of 31 delegates are at stake in Montana and South Dakota.
And about 180 superdelegates - party elders 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.
"There are a lot of superdelegates who are waiting for the last couple of contests, but I think that they are going to be making decisions fairly quickly after that," Obama told reporters in Michigan.
"My sense is that between Tuesday and Wednesday that we've got a good chance of getting the number that we need to win the nomination."
On Tuesday US representative James Clyburn, the highest ranking African-American in the US Congress, became the latest superdelegate to endorse Obama.
Obama has picked up 7.5 delegates on Tuesday alone.
Rob Reynolds, Al Jazeera's Washington correspondent, says 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".
On Monday Clinton said she would be making her case to superdelegates that she is the strongest candidate to beat John McCain, the presumptive Republican candidate, in November.
"The decision will fall to the delegates empowered to vote at the Democratic convention. I will be spending the coming days making my case to those delegates," Clinton told supporters in Yankton, South Dakota.
"We have a very strong case to make that I am the best positioned to take back the White House and put this country on the right track," she said.
But Hassan Nemazee, a national co-chairman of Clinton's finance committee, said that if Obama succeeded in reaching the delegate threshold on Tuesday, there seemed to be little reason to continue her candidacy.
"If one candidate has the requisite number of delegates, both pledged and super, it makes it far more difficult to make the credible argument that she stay in on the chance that some superdelegates might change their mind and endorse her later," he said.