Recognizing that a successful 2004 presidential challenge requires support from the middle, Senate Democrats desiring the privilege of squaring off against President Bush sought support from an influential group of House moderates Wednesday.

Sen. Bob Graham (D-Fla.) set the tone for the meetings when he told the New Democratic Coalition, which numbers 74 members, that “George Bush is vulnerable in 2004.”

“His economic plan is unlikely to stimulate the economy, will increase the debt and is unfair to the American people,” the not-yet-declared candidate said afterward.

And so it went as Sens. John Kerry (Mass.), Joe Lieberman (Conn.) and John Edwards (N.C.) paraded in at half-hour intervals, each telling the House Members why he and he alone could topple the popular president.

Noticeably absent from the gathering were Rep. Richard Gephadt (Mo.) and former Vermont Gov. Howard Dean, both of whom were invited but unable to attend. According to a spokeswoman, Gephardt had a “prior engagement” out of town.

All of the candidates will be invited back for a second session in which each will be allotted an hour with the group.

“I’m a charter New Democrat,” Lieberman said, playing up his ties to the group. “I feel very strongly that I can win in ’04 and that we will only be victorious with a New Democrat who is strong on security issues, has strong values and is willing to separate from his party when necessary.”

And while the rhetoric behind the closed-doors was undoubtedly high, House participants were reluctant to commit to a candidate.

Going into the meeting, Rep. Jim Moran (Va.) told Lieberman: “I was just saying nice things about you.”

To which Lieberman responded: “Keep talking!”

But on the way out, despite professing his personal fondness for former Vice President Al Gore’s 2000 running mate, Moran would not line up behind him just yet, saying he is not ready to adopt the Senator’s hawkish views on Iraq.

As Edwards, the last and youngest candidate to make his pitch to the New Democrats, left the meeting, he said he told his colleagues “how I plan to win and beat George Bush.”

Of course, he must beat at least five other Democrats — and counting — before he gets that chance.