As Democratic leaders try to finesse their way beyond a bitter primary, the two heavyweights stirring the pot met Thursday night in secret -- but whether anything was settled is anybody's guess.

Most of what Barack Obama and Hillary Clinton talked about stayed private in the hours after their meeting, and only a short, cryptic campaign message gave any hint of what happened behind closed doors.

"Senator Clinton and Senator Obama met tonight and had a productive discussion about the important work that needs to be done to succeed in November," their campaigns said in a joint statement released late in the evening.

But Clinton supporters swiftly re-launched speculation that she's being considered as Obama's running mate, just hours after the campaign declared she's not seeking the No. 2 spot.

After all, Clinton met in the middle of the night at the Washington, D.C., domicile of one of Clinton's biggest supporters, California Sen. Dianne Feinstein. And Feinstein was promoting the "dream ticket" concept the next morning.

"She brings constituencies ... women, elderly, working people, certain states that are critical to win," Feinstein told FOX News Friday when asked about the upsides of an Obama-Clinton pairing.

Feinstein said she set up Clinton and Obama Thursday night by her fireplace, and then went upstairs to give them privacy. She said she originally offered up her home to the candidates for the meeting, and Clinton took her up on the offer.

VoteBoth.com, a Web site founded by former Clinton aides and dedicated to uniting the rivals, said, "We are thrilled to hear that Senators Obama and Clinton are meeting already. We hope that Thursday night's conversation is a step toward an Obama-Clinton ticket that will unify the party."

New York Sen. Charles Schumer, a top Clinton supporter, said Friday that the meeting was not about the vice presidency, but that Clinton would accept the No. 2 spot.

"She has said if Senator Obama should want her to be vice president and thinks it would be best for the ticket, she will serve, she will accept that. But on the other hand, if he chooses someone else she will work just as hard for the party in November," Schumer told ABC's "Good Morning America."

An Obama spokesman elaborated on the overnight meeting, a little.

"Senator Obama and Senator Clinton did have occasion to meet this evening," said Obama spokesman Robert Gibbs. "It's the end of the primary process. They wanted to talk about bringing these campaigns together in unity."

Gibbs would not say where the former rivals met, except that it was not at Clinton's home in Washington.

Democratic strategist Bob Beckel, a FOX News contributor, said the meeting makes sense.

"Barack Obama needs to put the Hillary Clinton issue behind him. He's got to now start to concentrate on John McCain. ... For Obama to take a few extra hours and stay in the Washington area and to meet with her I think probably smooths a lot of things over prior to her speech on Saturday," Beckel said.

Clinton plans to publicly bow out of the race and endorse Obama Saturday, in an event party leaders hope will heal the lingering animus among Democrats.

Clinton said in an e-mail to supporters Wednesday that: "On Saturday, I will extend my congratulations to Senator Obama and my support for his candidacy. This has been a long and hard-fought campaign. ... I have said throughout the campaign that I would strongly support Senator Obama if he were the Democratic Party's nominee, and I intend to deliver on that promise."

Many Clinton supporters think the best way to unite the party is to unite the candidates.

But in a statement Thursday evening, the Clinton campaign stressed that the running-mate decision is up to Obama, and that Clinton "is not seeking the vice presidency."

Obama has indicated he intends to take his time making a decision. On Wednesday, he named a three-person vice presidential vetting team that included Caroline Kennedy, daughter of the late President Kennedy.

Majority Leader Harry Reid, who had remained uncommitted throughout the primaries, also endorsed Obama Friday.

Obama is "a once-in-a-generation leader who connects with the hopes and dreams of the American people and will deliver the long-overdue change that our country desperately needs," the Nevada Democrat said in a statement.

FOX News' Major Garrett and Trish Turner and The Associated Press contributed to this report.