The Senate’s top two leaders are making a last-ditch effort to avoid the across-the-board tax increases scheduled to take effect Tuesday.

After a White House meeting Friday afternoon, Majority Leader Harry Reid and Minority Leader Mitch McConnell said they would try to negotiate a bipartisan agreement by Sunday. “I’m hopeful and optimistic,” McConnell, R-Ky., said on the Senate floor.

The GOP leader’s upbeat assessment contrasted sharply with the partisan rancor that has accompanied the standoff on taxes and spending. President Barack Obama said at his own press briefing that he is “modestly optimistic.”

On the Senate floor after the meeting, McConnell said that he, Reid and the White House were “engaged in discussions . . . in the hopes that we can come forward as early as Sunday and have a recommendation that I can make to my conference and the majority leader can make to his conference.”

“We’ll be working hard to see if we can try to get there in the next 24 hours,” the minority leader said.

Later, Reid issued a statement casting things a little differently, saying that he is writing a bill for Senate action “by Monday” and that he would welcome any “good-faith proposals” from McConnell.

There were strong signs Friday that any pact would be limited in scope, addressing tax issues but not the $109 billion in spending cuts due to start taking effect on Jan. 2 or an increase in the government’s borrowing limit.

Obama said that if Reid and McConnell are unable to come up with something that both the Senate and House will pass, he has asked Reid to bring to the Senate floor a “basic package” representing a White House proposal.

Senators were advised at their caucus lunches to expect votes on Monday, which is New Year’s Eve. Even if leaders can strike a deal, it probably will require unanimous consent to proceed quickly, and that would leave the legislation vulnerable to an objection by any senator. That means the fate of the plan will be largely in the hands of McConnell, who would need to ensure that the proposal could move without a filibuster.

Reid, who said the Senate would hold votes Sunday on unrelated measures, expressed cautious optimism about finding a solution. “I’m going to do everything I can,” he said. “I’m confident Senator McConnell will do the same. But everybody — whatever we come up with is going to be imperfect. Some people aren’t going to like it. Some people will like it less.”

Any agreement in the Senate could face a difficult path in the House, where Speaker John A. Boehner, R-Ohio was unable last week to muster the votes for his own fiscal cliff package.

A Republican source briefed on Friday’s meeting said Boehner remained adamant that he would not support a plan to turn off the automatic spending cuts without alternative spending cuts to replace it.

Boehner did not speak with reporters on Friday, but an aide reiterated that the House has passed bills to head off the fiscal cliff issues. “The Speaker told the president that if the Senate amends the House-passed legislation and sends back a plan, the House will consider it — either by accepting or amending. The group agreed that the next step should be the Senate taking bipartisan action.”

For a week, Boehner has been calling on the Senate to provide the House with something to vote on. If things work out according to Reid and McConnell’s plans, that might happen.

Should a bipartisan agreement remain out of reach, the president wants a vote on his last fiscal cliff offer to congressional leaders a week ago — a plan that would raise some $80 billion in revenue in the next year by extending lower tax rates for all but the top two income tax brackets. That revenue would be partly offset by extending expiring long-term unemployment benefits.

The president’s proposal would protect everyone making $250,000 or less from a tax increase and would prevent 2 million unemployed from losing benefits, among other provisions.

The White House did not include an extension of the payroll tax cut, signaling that it would allow that tax to rise to 6.2 percent from the current 4.2 percent. That is worth $95 billion in revenue, according to the Tax Policy Center.

The proposal would extend the alternative minimum tax “patch,” continuing what has become an annual Congressional ritual and saving some 28 million households from being subject to higher taxes, according to the Tax Policy Center, and amounting to some $86 billion in tax savings for taxpayers.

Friday’s meeting came as some senators said they were working on their own 11th-hour bid, which would extend tax cuts for annual incomes up to $400,000.

Steven T. Dennis and Paul M. Krawzak contributed to this story.