Updated: 4:57 p.m.

The Senate could begin debate on President Barack Obama’s two-year extension of the Bush-era tax cuts as soon as Thursday, Majority Leader Harry Reid (D-Nev.) said Wednesday.

If Democrats are unable to invoke cloture during this evening’s vote on the defense authorization bill, lawmakers could begin debate Thursday morning.

Following a closed-door meeting Wednesday to discuss the tax cut deal, Reid told reporters, “I hope we can be on that in a day or two,” which could set the stage for a final vote this weekend or early next week.

Even if cloture is invoked on the defense bill Wednesday night, the chamber could take up the tax bill as soon as Friday or Saturday, Democratic aides said.

Beyond that, however, the road forward on the tax bill remains unclear. Sens. Bernie Sanders (I-Vt.) and Jim DeMint (R-S.C.) are both committed to filibustering the tax bill, and significant numbers of Democrats in both chambers remain at best skeptical about the agreement.

Senate Budget Chairman Kent Conrad announced Wednesday that, with some changes, he could eventually support the tax cut deal

“I think the President did the right thing in compromising with Republican leadership,” the North Dakota Democrat said in a statement. “The hard reality is this economy remains mired in unemployment and underemployment that affects one in every six Americans.”

Conrad was particularly concerned that the deal’s provision on the estate tax “goes too far.” It would set a lower maximum tax rate and a higher exemption than had been scheduled for 2011.