About 180 superdelegates - party elders who can choose to back either candidate at August's convention in 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.
"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." Clinton defiant
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.
Her husband and former president Bill Clinton 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".
She 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.Delegate threshold
But Hassan Nemazee, a national co-chairman of Clinton's finance committee, said that if Obama succeeded in reaching the delegate threshold on Tuesday, their 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.
Obama picked up 4.5 superdelegate votes on Monday, while Clinton picked up two.
James Clyburn, a hugely influential African-American leader and major force in Congress, was also expected to throw his support behind the Illinois senator on Tuesday.