It was only hours after Mayor Michael R. Bloomberg finally made it clear he was not running for president that his chief political guru, Kevin Sheekey, publicly floated the notion that he would be the perfect running mate for Senator Barack Obama.

So Mr. Bloomberg’s highly orchestrated introduction of Mr. Obama at a speech at the Cooper Union this morning quickly resuscitated speculation that the billionaire mayor could end up in the White House after all. (The Caucus has a roundup of the blog commentary.) But despite a few jokes and a stiff embrace, the men seemed nothing like two peas in the same political pod, destined to share a ticket.

In presenting Mr. Obama, Mr. Bloomberg pointedly noted that he has not endorsed a candidate, saying that now was the time for people to listen to the candidates’ ideas. Referring to the ostensibly private breakfast the two men shared at a Midtown coffee shop in November with a throng of news media pressed against the window, Mr. Bloomberg joked that it was his pleasure to introduce him, “and not just because he picked up the check.”

Mr. Obama had kind words for Mr. Bloomberg as well, saying that he had “demonstrated extraordinary leadership.” He joked: “The reason I bought breakfast is because I expected payback. I’m no dummy. The mayor was a cheap date that morning. There are some good steakhouses in this city.”

But during the speech, Mr. Bloomberg sat on the side near an exit without applauding, maintaining a stance of political neutrality by keeping his hands in his lap as Mr. Obama went after Senator Hillary Rodham Clinton, the Democrat from New York, and John McCain, a Republican Mr. Bloomberg counts as a friend.

Still, with recent meetings with Al Gore and Chuck Hagel, interest in Mr. Bloomberg’s future has been running high, at least among the political consultants and the media who wonder if Mr. Bloomberg could make a back-door run at the presidency by teaming up with Mr. Obama or Mr. McCain, or if a cabinet position, like Treasury Secretary, will be in the offing.

Stu Loeser, Mr. Bloomberg’s press secretary, sought to dampen the vice presidential notion, saying, “He’s trying to shape and influence the campaigns, he’s not trying to join one.”

Still, Mr. Bloomberg gained in another way, as Mr. Obama endorsed his congestion pricing proposal.

Michael Powell contributed reporting.