Mr. McConnell’s decision averted what had been shaping up as a remarkable collapse of those talks. House Democrats scheduled a Thursday vote on their own package of paid sick leave, enhanced unemployment insurance, free coronavirus testing and food aid, and Senate Republicans, who oppose that legislation, faced the prospect of leaving Washington having taken no action to address the widening crisis.

Instead, the House postponed its vote as the negotiations appeared to progress on Thursday, and Mr. McConnell said he, too, would allow time for a deal by bringing senators back to Washington next week.

Ms. Pelosi spoke eight times on the phone with Mr. Mnuchin on Thursday, even as President Trump attacked her on Twitter for refusing to embrace the huge payroll tax cut he has proposed, which has drawn bipartisan opposition.

“He had some suggestions,” Ms. Pelosi said of Mr. Mnuchin. “All very reasonable. I don’t think that any of them would prevent us from moving forward with the bill.”

But some major sticking points were hampering agreement, according to people familiar with the deliberations, who insisted on anonymity to describe them. Republicans balked at a sweeping proposal to provide paid sick leave, something Senate Republicans had already blocked when Democrats sought earlier in the week to bring up a separate bill. And Republicans were insisting on inserting language into the emergency package to ban federal funding for most abortions.