The day before Hillary Clinton racked up her emphatic—if mostly meaningless—victory in the West Virginia primary, Barack Obama’s chief strategist was in an Upper East Side apartment with dozens of political insiders, dealing with the new facts on the ground.

Surrounded by views of the East River and trays of salmon-filled pastries, sesame chicken and lobster dumplings, the strategist, David Axelrod, addressed a room of about 75 Democratic activists and fund-raisers that included a number of newly converted Clinton supporters.

“Obama’s the candidate,” said Sarah Kovner, a former Clinton administration official and longtime Clinton ally who was in attendance. “I’m a Democrat and I have been a Democrat for a long time and I want to win in November, and I don’t want to continue a fight that isn’t a fight anymore.”

Ms. Kovner said she had not yet made a contribution to the Obama campaign, but she added, “I certainly will, as will everyone I know.”

Ms. Kovner, an insider’s insider, was stating as fact what the Clinton campaign can’t yet say publicly—that the race for the nomination is over.

Interestingly, the Obama campaign and fund-raising apparatus is also taking care not to state the premise quite so clearly. In fact, with the nomination no longer in doubt—they have netted enough superdelegates in the past week to erase Mrs. Clinton’s gains from a lopsided West Virginia win—Mr. Obama’s top aides and strategists have stopped worrying entirely about whether Clinton supporters are going to come over, concentrating instead on the most elegant means of paving the way.

“It’s about thinking who is on the leadership for the women’s groups for Clinton, the young professionals for Clinton, making sure we understand we know who is on the leadership side of that and understanding who on our side knows those people to reach out to them at the appropriate time,” said one influential Obama bundler. “Thinking of what kind of events you might do to facilitate that integration, making sure the people on our side, up and down the leadership line, are welcoming and not carrying the slights that are felt on both sides.”

Certainly, that was a key part of Mr. Axelrod’s mission during his visit to Mrs. Clinton’s financial and political base—to remind her supporters of the inevitable, but also to make preparations for their eventual incorporation.

According to several guests at the $1,000-a-head fund-raiser, he talked about the campaign’s national voter-registration efforts reflecting its 50-state strategy; assured supporters that the campaign would respond quickly and convincingly to any coming Swift-Boat-style attacks; and recognized the difficulty members of the New York donor community had endured in supporting Mr. Obama early on.

He said the primary characteristic Mr. Obama would look for in a vice president was someone with whom he was extremely comfortable (which may not bode well for Mrs. Clinton’s chances). He said he had no idea when Mrs. Clinton would stop running, but pointed out the seemingly inexorable march of superdelegates to his candidate. He continued to address Mrs. Clinton with only the utmost respect, though, taking care to call her a formidable and worthy competitor.

According to several Obama bundlers speaking on background, their job at this point wasn’t to flip the Clinton donors, but to achieve a seamlessness in fund-raising operations between the primary and general elections.

“I am a bridge,” said Leonore Blitz, an Obama bundler and advocate for women in politics who attended the May 12 fund-raiser. She said that the guiding principle at the Obama campaign was not to disturb Mrs. Clinton’s more significant backers until they were ready. “These top Hillary supporters and raisers, I haven’t approached them, and I’m sympathetic,” she said. “But the people I know who have maybe written a check to her? Yes, I have called them. And they have called me.”

“When the flag gets raised, everybody has some place to come and it’s organized,” said one of the donors. “It’s not like, ‘We’ll be back to you in two weeks to tell you how you could start participating.’ Because that alone could be off-putting.”

Still, some of Mrs. Clinton’s top donors are nervous.

There is a feeling among them that the small window of time to raise money between Mrs. Clinton’s probable withdrawal in June and the convention poses a challenge for even the most accomplished bundlers to make their mark and earn the recognition and appreciation from the Obama campaign’s leadership.

Complicating matters is the lack of friends the Clinton donors have in the Obama camp. When the campaigns of Wesley Clark and Howard Dean collapsed ahead of the 2004 election, the party’s big-name fund-raisers found a home, and prominent roles and titles, with the Kerry campaign, thanks in part to their relationships with Hassan Nemazee, who was then Mr. Kerry’s national finance chair.

“What I can say is, if I were the national finance chair of the presumed nominee, I would do everything in my power to welcome supporters of the opposing side into the campaign as much as possible,” said Mr. Nemazee in an interview this week.

But the Obama campaign’s financial structure has an open architecture that frowns upon titles and is anathema to the “Hillraiser” system within which the old guard thrived.

The best hope for Mrs. Clinton’s biggest fund-raisers of finding a warm welcome in the Obama campaign may be Orin Kramer, a New Jersey-based financier and veteran of the Carter administration. Mr. Kramer was pretty much the only super-elite Democratic fund-raiser in the New York area who signed on early with Mr. Obama. (Virtually all the rest were with Mrs. Clinton.)

Other than Mr. Kramer, the Obama campaign has few fund-raisers in leadership positions with long-standing relationships to Clinton donors. For example, Penny Pritzker, the Obama campaign’s national finance chair, is a largely foreign figure to the New York, Los Angeles and San Francisco fund-raising circuit that has dominated Democratic politics for the past 20 years.

As a result, Mrs. Clinton biggest bundlers suspect that there’s no way in at this point: that even if they succeed in raising significant sums—as much $250,000 each—for Mr. Obama, their efforts will not lead to anything like the access and input to the candidate and senior staff they enjoyed with the Clinton campaign.

They may be right.

“I’m not sure they are ever coming over because there has been such a loss of face,” said one Obama donor. “I’m not sure they love Hillary so much, but they are sure used to getting their own way. And not getting their own way just burns them.”