Clyburn, the highest ranking African American in Congress, said Clinton dropping out of the race on Tuesday would be "just a technicality."
When asked if Obama should pick her as vice president, he said "I would not be insulted by that."
Several of Clinton's supporters have called for a joint ticket.
"There's a growing movement that's becoming almost bigger than Hillary Clinton to make sure they are on the ticket together," said Mark Aronchick, a Philadelphia lawyer and top Clinton fundraiser.
Obama told a small group of reporters Monday that he's offered to meet Clinton on her terms "once the dust settled" to discuss a unified strategy to win back the White House.
"The sooner we can bring the party together, the sooner we can focus on John McCain," he said.
Clinton prepared for a campaign rally Tuesday night in her home state of New York — a departure for the senator, who usually gathers supporters after a caucus or a primary in a state where she's been campaigning. Top donors have been invited to attend the rally.
A total of 31 delegates are at stake in the final primaries.
Obama is about 40 short of the 2,118 delegates needed to win the nomination. His campaign expects he'll lock it up by the end of the week with the help of superdelegates, the party leaders who are free to endorse any candidate.
Clinton's biggest supporter, her husband, hinted that the end could be near. "This may be the last day I'm ever involved in a campaign of this kind," former president Bill Clinton told voters in Milbank, S.D.
Hassan Nemazee, co-chairman of Hillary Clinton's finance team, said Clinton has not been specific about her endgame.
"I think she's a political realist … and will determine at the appropriate time what's best, not only for herself, but for the party," Nemazee said.
For the second day in a row, Obama tossed verbal bouquets at Clinton before a crowd of his supporters. He assured them that the Democratic rift will be repaired.
"She and I will be working together in November," Obama told a crowd of about 3,000 at the Troy High School gym.
Sunday, Clinton sounded a conciliatory note, saying she wanted "to recognize Sen. Obama and his supporters" as she heralded her win in the Puerto Rico primary.
Obama told a South Dakota crowd Sunday that Clinton will be "a great asset" in the general election campaign.
Political observers say Obama's shift in tone signals that he's trying to bring the contest to an amicable conclusion.
"He is reaching out to Hillary and trying to make it easier for her," said James Thurber, a political scientist at American University.
Kathy Bushkin Calvin, spokeswoman for Gary Hart's unsuccessful Democratic presidential campaign in 1984, said the stage needs to be set for the campaigns to find common ground.
"The goal is to find a way for face to be saved, for credit to be paid and for a platform to be created that supporters can stand on together," she said.
Ron Walters, a political scientist at the University of Maryland, said Obama's wooing of Clinton at this stage should not be viewed as a possibility that he is thinking of her as a running mate. The outreach is "designed to open the door as widely as possible to her constituency," Walters says, including white women who have not supported Obama in previous contests. "He's trying to convince them that he respects her."
Contributing: Fredreka Schouten; Kathy Kiely in Troy, Mich.; and the Associated Press