“After June 3, this is going to come to a conclusion,” Mr. McAuliffe said on NBC’s “Today” program.

Other Clinton allies said much the same thing, evidence of a consensus that Mrs. Clinton has four weeks to make her case but then should exit quickly if she has not turned the race around. Her campaign grappled with a number of impediments to fighting on, including a decline in fund-raising.

“I think she should complete the primary season, and then she has to re-evaluate and her supporters have to re-evaluate,” said Representative Nita M. Lowey of New York, a Clinton backer.

Mr. Obama made no public effort to pressure Mrs. Clinton from the race, and in interviews with CNN and NBC News he praised her as a formidable candidate who could not yet be counted out. But he said that he was likely to lock up a majority of the pledged delegates  those awarded by voting in the primary and caucus states  after the Kentucky and Oregon primaries on May 20, and that at that point he could declare victory.

While he was respectful to Mrs. Clinton, Mr. Obama seemed eager to challenge Mr. McCain. Asked on CNN about Mr. McCain’s recent statement that the radical Palestinian party Hamas, considered by the United States to be a terrorist organization, would favor Mr. Obama’s election, Mr. Obama said it was offensive and called it a smear.

“And so for him to toss out comments like that I think is an example of him losing his bearings as he pursues this nomination,” Mr. Obama said.