WASHINGTON — President Obama will meet with Congressional leaders on Friday, and House Republicans summoned lawmakers back for a Sunday session, in a last-ditch effort to avert a fiscal crisis brought on by automatic tax increases and spending cuts scheduled to hit next week.

Republicans expressed a flicker of hope Thursday that a deal could still be reached to at least avert most of the tax increases on Jan. 1, to prevent a sudden cut in payments to medical providers treating Medicare patients and to extend expiring unemployment benefits. But both parties’ leaders said time is running out.

“Here we are, five days from the New Year, and we might finally start talking,” said Senator Mitch McConnell of Kentucky, the Senate Republican leader.

The overriding emotion Thursday, as senators convened for a rare session between Christmas and New Year’s Day, appeared to be embarrassment. The continuing impasse “demonstrates a tremendous lack of courage here in Washington to address the issues that need to be addressed — at every level,” said Senator Bob Corker, Republican of Tennessee.