Senator Clinton, Mayor Giuliani, and Governor Pataki could have competition if they want Mayor Bloomberg’s endorsement for the White House.

While Mr. Bloomberg insists he’s not running, he lightheartedly suggested that one of his advisers, who’s been fueling the recent rash of “Bloomberg for President” stories, should seek the job himself.

Mr. Bloomberg deflected questions about why Deputy Mayor Kevin Sheekey has been keeping the story alive even as the mayor repeatedly denies aspirations for higher office.

“If my adviser is planning to run for president, I wish him all the best. He’s a very competent guy,” Mr. Bloomberg joked. “I would be happy to consider supporting him. I think he’d do a great job for this country. He could run as a Republican, a Democrat, or any other party he wants to. Personally, I am not running.”

In a recent interview with the New Republic, Mr. Sheekey said that if Senator McCain loses the GOP primary and the Democrats end up with a candidate they perceive as “unelectable” (i.e., Mrs. Clinton), there would be a “large segment of the American electorate that is looking for something different.”

Some say that’s a coded reference to Mr. Bloomberg, who would likely run as an Independent.

When asked whether he had told Mr. Sheekey to stop discussing the matter with reporters, Mr. Bloomberg quipped: “I’m shocked, shocked that you would think that Kevin Sheekey would encourage rumors. He is certainly not that kind of person. Somebody who worked for Daniel Patrick Moynihan would never do that.”

Mr. Bloomberg, a billionaire who could easily self-finance a national campaign, reportedly told guests at a fund-raiser earlier this month that anyone with presidential aspirations would deny it this long before an election, just as he’s been doing.

Mr. Sheekey seemed to rule out a run of his own. He said via e-mail: “Unfortunately, my candidacy would not come with his business success, political accomplishments … or money.