Yet it remains unclear who can stop the popular Mr. Bloomberg from changing the law and coasting to an election victory next year. The concept of a third term has broad support among City Council members, business leaders and newspaper editorial boards, all of whom Mr. Bloomberg has assiduously courted behind the scenes. On Wednesday, The New York Times published an editorial calling for term limits to be abolished. The New York Post and The Daily News have already endorsed Mr. Bloomberg’s bid for a third term.

Image Representative Anthony D. Weiner called the notion illegal. Credit... Librado Romero/The New York Times

And, according to a person who has advised the mayor in the past, he “does not care what it costs” to win a third term. In 2005, Mr. Bloomberg, a billionaire, spent more than $70 million and won nearly 60 percent of the vote.

Mr. Bloomberg was observing the Jewish New Year on Tuesday and his press aides did not return several e-mail messages and telephone calls.

During marathon telephone conversations in recent weeks, Mr. Bloomberg has debated with his friends and advisers the pros and cons of running again.

“This has been thoroughly thought out by the mayor,” said the person who has advised the mayor in the past. “It’s not like he just woke up one morning and decided to do it.”

With his decision, Mr. Bloomberg is overruling the advice of his top three assistants at City Hall  Deputy Mayors Edward Skyler, Patricia E. Harris and Kevin Sheekey  who have expressed opposition to a third term.

Those aides have told the mayor  at times forcefully  that any campaign to challenge the term-limits law would look like an end run around voters, and could sully his legacy as a reform-minded outsider. Others have told the mayor that they may not remain for a full third term.