WASHINGTON – Senate leaders said Friday they will spend the weekend attempting to patch together a last-minute deal to avoid the fiscal cliff, pledging another bipartisan effort despite growing doubt about the ability of the White House and Congress to reach agreement.

“I’m hopeful and optimistic,’’ said Senator Mitch McConnell, the Republican minority leader, in a marked departure from the partisan jibes and finger-pointing that have marked most of the public dialogue in recent days.

“We are going to be working hard to see if we can get there in the next 24 hours,’’ McConnell said.

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McConnell and Senate Majority Leader Harry Reid, the Senate’s top Democrat, joined their House counterparts at the White House for an afternoon meeting where they agreed to seek an agreement that could pass the Senate and then go to the House for a vote.

President Obama, appearing shortly afterward in the White House briefing room, said that if the Senate leaders do not produce a compromise plan before the Jan. 1 deadline, he would ask Reid to seek an “up or down vote’’ on a Senate bill to extend tax cuts for the middle class and continue unemployment benefits for 2 million people. Going over the fiscal cliff, the president said, would harm the economic recovery.

“The American people are not going to have any patience for a politically self-inflicted wound to our economy. Not right now,’’ he said. “Already you are seeing businesses and consumers start to hold back because of the dysfunction they see in Washington.’’

Obama plans to appear on NBC-TV’s “Meet the Press” on Sunday to make his case for passing legislation before the deadline, the network announced Friday night.

After a week of recriminations and stalled talks, Friday’s cautiously positive remarks following the meeting marked yet another twist in the dramatic showdown over $500 billion in automatic tax increases and spending cuts due to take effect on Jan. 1. Americans will likely know whether it was merely another case of false hopes, or will lead to an actual breakthrough, by Sunday.

No details of the parameters of their discussions were released. Senate leaders have discussed working toward a smaller-scale deal to avert the automatic income tax increases for a majority of Americans, while also dealing with other emergency issues like extending unemployment benefits and fending off deep cuts in Medicare reimbursements for doctors and hospitals.

“Whatever we come up with, some people aren’t going to like it,’’ Reid declared after the meeting, joining McConnell on the Senate floor to provide an update. “We’ve had a constructive meeting. We certainly hope that something positive will come from that.’’

Republican Speaker John Boehner is prepared to bring to the House floor the fruit of a McConnell and Reid compromise if it passes the Senate, according to a statement issued by the speaker’s office late Friday afternoon. At that point, the House would either accept the bill or amend it, the statement said.

That indicates Boehner would be willing to present a deal that might rely on a coalition of Democrats and Republicans for passage – in contrast to his failed efforts last week to win passage, solely with GOP support, of a bill extending tax cuts for everyone except millionaires.

House Minority Leader Nancy Pelosi also attended the White House meeting Friday, along with Vice President Joe Biden and Treasury Secretary Timothy Geithner. The meeting began at 3 p.m. in an atmosphere of increasing tension and anxiety about the Jan. 1 fiscal cliff deadline. Economists predict that the combination of tax increases and spending cuts due to be triggered at the fiscal cliff would likely trigger a deficit during 2013.

Boehner has summoned House members to arrive in Washington on Sunday night in the event there is something for representatives to vote on Monday, the last day before the New Year’s Eve deadline.

Business and political leaders around the country have been watching the sputtering negotiations with a growing sense of dread.

In Massachusetts—where healthcare is a mainstay of the economy—healthcare executives are fighting two sets of Medicare cuts. In the automatic ``sequester’’ spending cuts that were put into place as part of an August 2011 deal to raise the debt ceiling, Medicare spending is scheduled to be reduced by 2 percent overall, or about $11.1 billion in 2013.

The state’s physicians and doctors have their eye on another major spending issue that is said to be part of Obama’s proposal: the need for Congress to override a healthcare budget control put in place years ago and tweak the Medicare reimbursement formula for doctor and hospitals. If the so-called ``doc fix’’ is not passed, as has been the usual practice, it would result in a 27 percent cut to Medicare payments effective on Jan. 1.

Over the past decade, in what has become a tired dance, hospitals and doctors repeatedly have warned that the cuts would decimate Medicare and hurt patients. Congress has always agreed. With the ``doc-fix’’ bound up in the fiscal cliff negotiations, some doctors and hospitals in Massachusetts fear they will not be as lucky this year.

“This is very real,” said Lynn Nicholas, chief executive of the Massachusetts Hospital Association. “It’s not just political rhetoric this time.”

A state health care law passed this summer puts pressure on hospitals and doctors to save money. With similar measures in the Affordable Care Act, the looming Medicare cuts, and more expected to funding for medical research, “there is no relief on the federal front,” she said.

If the cuts stand, Nicholas said, the state could see in an increase in health care layoffs and hospitals will have to spend their cash reserves as they figure out how to adjust to the drop in revenue. Older doctors might consider retiring and others will close their private practices for a job on a hospital payroll, said Dr. Richard Aghababian, president of the Massachusetts Medical Society.

“This is not rocket science,” he said. “A 27 percent cut is going to have a huge impact.”

But the health care industry should be pushed to reduce wasteful spending, said Dr. Gene Lindsey, chief executive of Atrius Health and Harvard Vanguard Medical Associates. A 27 percent cut in Medicare would be too much all at once, he said—a cleaver versus a scalpel. Some hospitals are figuring out how to do better by, for example, putting processes in place to avoid repeat diagnostic tests. While those changes take time, he said, a small cut to Medicare, something around 2 percent, could help those efforts along.

“Hopefully what will happen now is just enough to get people’s attention and not enough to cause damage,” he said.