Under threat of a rare all-night session, Sen. Jim Bunning (R-Ky.) and Republican and Democratic Senate leaders appear close to a deal to end the Kentucky Republican’s blockade of a one-month extension to unemployment benefits, Democratic and Republican aides said.

Although an agreement has yet to be finalized, the proposal under consideration would allow Bunning one vote on an amendment to cover the bill’s $10 billion price tag — essentially the same agreement that Majority Leader Harry Reid (D-Nev.) and Minority Leader Mitch McConnell (R-Ky.) reached before Bunning rejected it Thursday and launched his standoff.

Bunning would also be given two votes on amendments to pay for a larger, one-year extension of unemployment and health insurance benefits and other programs later this week, the aides said.

A vote on the short-term extension could occur as soon as 6:30 p.m. Tuesday.