Senate Majority Leader Harry Reid (D-Nev.) and Minority Leader Mitch McConnell (R-Ky.) said on Wednesday that they would try to work out a Thursday vote on a House measure that would extend unemployment insurance, a move that would signal the end of the week’s lame-duck session.

On Wednesday evening, Reid filed cloture on a bill that would extend unemployment insurance benefits after he failed to get unanimous consent for an economic stimulus package that included infrastructure funding, state aid for Medicaid, and a loan for the ailing automotive industry. Republicans objected to all three bills.

The cloture petition guarantees there will be a vote on unemployment benefits by Friday, but Reid and McConnell are hoping to move that vote up by a day, and said they would go back to their colleagues to work out an agreement that would allow a Thursday vote.

If the Senate succeeds in its Thursday vote, there is no reason for the House to come back to town because it has already passed the measure.

One Democratic leadership aide said a Thursday vote is very likely now that Reid is pushing only for the unemployment insurance, which has significant Republican support.

“They want to leave. A vote could happen as early as Thursday,” the aide said.

Reid tried all week to persuade Republicans to give the Big Three American auto-makers a $25 billion loan that would help with their more pressing needs, but ran up against President George W. Bush and Senate Republicans, who argued that those companies need to restructure themselves instead.