Bloomberg, a Democrat-turned-Republican-turned-independent, is due to leave office at the end of 2009. He announced last week that he would push for City Council legislation to give city officials the option of running for a third term. Voters set the current limit - two consecutive four-year terms - in a 1993 referendum and reaffirmed it three years later.
A City Council member opposed to Bloomberg's plan has introduced a competing proposal that would require a voter referendum on any change in the term-limit law.
Quinn said at a City Hall news conference that she had decided to support Bloomberg's proposal "after careful conversations with my colleagues and careful deliberation."
She said Bloomberg had promised no quid pro quo in return for her support, nor had she threatened council members with the loss of perks and committee chairmanships if they opposed Bloomberg's bill.
Elected officials who oppose Bloomberg's plan said any change in the term-limit law should go before the voters, not be left up to the 51-member council.
"There is a right way and a wrong way to change term limits," said Rep. Anthony Weiner, another likely mayoral candidate. "And the mayor and the speaker have chosen the wrong way...The right way is to return to the citizens of the City of New York and ask them for their decision about something they put in place."
City Councilman David Weprin noted that voters have backed term limits twice.
"The voters spoke," he said, and any change "should only go back to the voters."
Former Public Advocate Mark Green, who joined Weprin, Weiner and others on the steps of City Hall before Quinn's announcement, said Bloomberg was letting his desire to remain in office cloud his judgment.
"It's profoundly unethical to try and win an office when you knew at the start that there were term limits of two terms, when for seven and a half years you agreed with that, saying it was essential ... and then, at the 11th hour, you do a 180-degree flip because you want to cling to office," Green said.
Bloomberg vetoed proposed changes to the term-limit law in 2002 but has said he had a change of heart amid the Wall Street turmoil, which has dealt a heavy blow to the city's economy.
"The mayor very much appreciates Speaker Quinn's leadership in helping provide New Yorkers with steady hands on the rudder in the tough times we face," spokesman Stu Loeser said.
The council is set to hold a hearing on changing the term-limit law Thursday.