Senator Mitch McConnell of Kentucky, the Republican leader, said Sunday morning that he was “very close” to recommending to his members that they sign on to a debt deal with President Obama and the Democrats.

Speaking on the CNN program “State of the Union,” Mr. McConnell said the deal included as much as $3 trillion in cuts over the next 10 years, with much of that decided later this year by a joint congressional committee.

“What conservatives want to do is cut spending,” he said. “We’ve come a long way. This agreement is likely to encompass up to $3 trillion is spending cuts.”

In addition, Mr. McConnell said the agreement would allow votes in Congress on a balanced budget amendment.

Senator Charles E. Schumer of New York, a top Democrat in the Senate, cautioned on the same program that “there is no final agreement. No one has signed off on a final agreement.”

But he indicated the he and others were expressing optimism because of the ongoing discussions between the congressional leaders.

“If there is a word that would sum up the mood it would be relief,” he said. “Relief that we won’t default.”