Officials from the transit authority and the Transport Workers Union Local 100 have agreed to a new round of talks scheduled for Saturday, more than 24 hours after a contract deadline passed.

The union has agreed to postpone a strike of the city’s subways and buses until 12:01 a.m. Tuesday, but union workers at two private bus lines in Queens that were recently acquired by the city could walk off the job as early as Monday morning if a contract settlement is not reached.

The chairman of the Metropolitan Transportation Authority, Peter Kalikow, presented Friday what he called a “final offer” to the transit union representing 33,700 bus and subway workers, whose three-year contract expired at 12:01 a.m. Friday.

Just after midday Mr. Kalikow, who had joined the late night negotiations an hour before the union contract expired, addressed reporters on a third floor conference room at the Grand Hyatt hotel in midtown Manhattan where he laid out the details of an offer he said was “at the limits of our financial ability.”

Mr. Kalikow said the MTA would increase workers’ wages by 3% a year over a 36-month contract. The authority had an earlier offer on the table of two 3% raises over a 27-month contract. The union, the Transport Workers Union Local 100, rejected that offer in part because it would expire in March, when a strike would be less calamitous to the city’s businesses.

The union had been asking for wage increases of 8% a year over three years, but indicated Thursday that they were willing to bring that demand down in exchange for concessions from the authority over pension and health benefits, where the greatest impasses between the two sides exist.

The union has resisted efforts by the MTA to raise the retirement age for new workers from 55 to 62. Agreeing to do so, the president of Local 100 of the Transport Workers Union, Roger Toussaint, said, would be like “selling out the unborn,” referring to the benefits package of future employees.

On Thursday Mr. Kalikow, using the union’s catchphrase as his own, said that the union would have to recognize that rising health care and pension costs made the current benefits plan unsustainable.

“It might be easy to ignore this fact but that would be a disservice to our riders in the city, now and still unborn,” Mr. Kalikow said.

Pension costs for the MTA have tripled in the past three years to $450 million a year..

The pension offer made by the MTA did represent a change in its previous stance: the authority now says workers can qualify for a pension equal to half their salary at age 62 with 25 years of experience, down from 30. Sounding frustrated at what he believed to be spin coming from the union, Mr. Kalikow said the MTA has never during the contract negotiations attempted to change the benefits of current employees.

“I must reiterate that no existing MTA employee will have their health or pension affected by one penny,” Mr. Kalikow said.

Workers would remain eligible for a pension equal to half their pay after 25 years. The age of retirement would remain 55 years, Mr. Kalikow said.

The authority also lowered its demand that new employees contribute a portion of their wages to their health plan. The MTA had sought a 2% contribution but lowered it to 1%.

New talks have been scheduled for Saturday, though the union has yet to respond to the authority’s “final offer.” Mr. Kalikow and his top negotiator, Gary Dellaverson, told reporters Friday that they were ready to resume talks, but that the “bucket was full,” Mr. Dellaverson said, meaning no more concessions that would increase the MTA’s financial outlay could be made.

If the union is unable to agree to the offer, Mr. Kalikow said again that both sides should enter into binding arbitration, an option outlined under the state’s Taylor laws, the same laws that prohibit state civil employees form striking.

On Thursday, Mr. Toussaint rejected a similar claim made by Mr. Kalikow, in a series of exchanges that escalated tensions and the possibility of a strike.

Mr. Toussaint said the union would not enter into arbitration.

If the approximately 700 workers from the private bus lines, Jamaica and Triboro bus lines, which serve outlying parts of the borough, walk off the job, more than 55,000 riders will have to scramble for a ride to work Monday morning.

Mayor Bloomberg said Friday that commuter vans licensed by the city will be allowed to pick up riders along the buses’ routes and charge them $5 a person. Yellow taxis will be allowed to charge up to $10 a person.

The union believes that these workers, whose companies were acquired by the city and whose operations have been taken over by the authority, are not subject to the state’s Taylor laws, which dock each worker two days pay for every day they strike, because the official takeover begins next month.

Much of Thursday night had a scripted quality to it, with both sides holding news conferences to coincide with the 6 p.m. and 11 p.m. television newscasts. Talks began in earnest when Mr. Kalikow later sat at the negotiating table with Mr. Toussaint.

But by then, with a deadline minutes away, tensions were at a boiling point.

Earlier at 10:30 p.m. Thursday Mr. Toussaint stood with other city union leaders, including Pat Lynch of the patrolman’s union;mRandi Weingarten of the teachers, Dennis Rivera of 1199, and Brian McLaughlin of the Central Labor Council, attempting to portray the contract negotiations as a struggle for working people across the city.

At 11:25, the MTA’s top negotiator, Gary Dellaverson, angrily rebuffed that charge. “To begin to recharacterize the negotiations as some broad-based attack on the labor movement, on working people in the city, is simply wrong and it doesn’t help us reach an agreement,” he said.

In order to implement the city’s extensive contingency plan, the mayor said he would declare a state of emergency if a strike occurred. Meanwhile, the transit authority ordered supervisors to posts throughout the subway system where they would monitor MTA property against vandalism.

Mr. Kalikow had repeatedly said that reducing soaring pension and health care costs was the authority’s primary concern. The MTA board on Wednesday voted to spend $450 million of its $1.04 billion surplus to pay down its pension liabilities, which total $2.2 billion.

Governor Pataki said Thursday he would not get involved in the state authority’s negotiations, but he reiterated that a strike is illegal. “If they break the law, they will suffer the consequences of breaking the law,” he said.

Complicating matters, tensions within the union that had been simmering all week boiled over into physical confrontations between members, with dissident members accusing Mr. Toussaint of freezing them out of negotiations.

Union members angry at Mr. Toussaint for delivering what they believe to have been a bad contract in 2002 said they had been kept out of negotiations yesterday. A vice president for the union, John Mooney said some members, including himself, were roughed up by security guards who were part of Mr. Toussaint’s detail.

“It’s an outrage in these times,” Mr. Mooney, said. “If he can intimidate a vice president then he can intimidate the rest of the membership, and that’s what he wants to do.”

The union says workers deserve a better contract than in 2002, when they each received a 6% raise, plus a $1000 bonus, over three years. The presence of the surplus has only heightened the expectations and the potential for internal conflict.

“Dissidents only function when they respond to a discontent in the membership,” the director of the National Center for the Study of Collective Bargaining in Higher Education and the Professions at Hunter College, Richard Boris, said. “And I think the MTA workers want better than they got in the last contract.”

Mr. Boris said both sides would prefer not to strike, but that a strike could serve to solidify Mr. Toussaint’s constituency behind him, just as it would rally the public behind Mayor Bloomberg.

The last transit strike was in 1980 and lasted 11 days.