Her stance on the issue has been loudly booed at public forums, picked over in newspaper editorials and derided at weekly news conferences by a leading rival in the race for mayor.

Now, after three years of firm resistance, Speaker Christine C. Quinn of the City Council is changing course, entering serious negotiations on a bill requiring businesses to provide compensated sick days for employees, according to those involved.

The talks, which represent a major breakthrough for advocates of the legislation, follow weeks of intense and sustained pressure from liberal groups [including Make the Road New York] and Ms. Quinn’s competitors in the campaign, who have excoriated her for opposing the measure.

The discussions on the bill are moving rapidly, but remain precarious. Both sides cautioned that they could end without a deal.

“My understanding is there’s a real effort to find a workable compromise this week,” said Kathryn S. Wylde, a Quinn ally on the issue and head of the New York City Partnership, a business group that has objected to the mandate.

As speaker, Ms. Quinn has exercised her power to block a vote on the legislation, despite widespread support for it in the Council.

Ms. Quinn, a former housing activist who has strengthened her ties to business in her rise to the speaker’s office, has argued that the economy is too fragile to impose any new costs on small businesses.

Supporters say that legally required time off for those who are ill is a basic right, without which workers must choose between their health and their paychecks — and that they could sicken strangers by choosing the latter.

At issue in the negotiations are the size of the companies that would be affected, the mechanism by which it would be enforced and when the mandate would take effect.

The bill under consideration by the Council, which Ms. Quinn opposes, calls for all companies with five or more employees to offer paid sick leave. In the talks, her office has pushed for a higher number, around 20 to 30 workers, according to the people told of the discussions. Those people spoke on condition of anonymity, citing the confidentiality of the negotiations. Ms. Quinn’s office declined to comment on the state of the negotiations.

There appears to be a split among advocates over what would constitute an acceptable deal. As talks intensified on Tuesday, a major union representing health care workers, 1199, and one of Ms. Quinn’s leading opponents in the Democratic primary for mayor, Public Advocate Bill de Blasio, suggested that a compromise bill along the lines of what Ms. Quinn apparently wanted would exclude too many New Yorkers.

Mr. de Blasio said that a cutoff of 20 employees would leave about 384,000 city residents without legally mandated paid sick time.

“The only thing that is worse for these New Yorkers than waiting for a bill is waiting three years for a watered down bill that will leave out huge numbers of New Yorkers,” he said.

The details of laws on paid sick leave, which have galvanized liberal groups across the country, vary. In Philadelphia and in Portland, Ore., which have passed similar bills recently, the requirements would affect businesses with at least six workers. But a less stringent Connecticut law applies only to employers with 50 or more workers.

Ms. Wylde warned that a five-employee threshold could be “extraordinarily damaging” to the city’s economy.

“A five-employee business,” she said, “is simply not of a size to afford any extra costs.”

The bill’s fate has taken on an outsize role in the Democratic primary. All four of Ms. Quinn’s opponents have called for her to allow a vote, at times confronting her at candidate forums to make her position appear out of the party’s mainstream.

But it has put her in line with Mayor Michael R. Bloomberg, who has pledged to veto the bill as proposed.

Ms. Quinn’s decision to enter negotiations was prompted by a few factors, including a new openness among advocates for negotiation and the prominent role in the talks of Local 32BJ of the Service Employees International Union, a private sector labor group that has found ways to work with her in the past.

At the same time, Ms. Quinn and her team were unsettled by a plan, hatched by council members, to circumvent her by forcing a floor vote on the bill, a maneuver never tried during her tenure.

Such a move, if successful, could undermine Ms. Quinn’s authority as speaker and create an awkward moment for her with the approval of a cherished liberal cause over her objections.

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