Senate Minority Leader Harry Reid (D-Nev.) says Democrats are willing to debate immigration only after the Department of Homeland Security is fully funded. (Reuters)

This post has been updated 

The Senate voted Wednesday to move ahead with a bill to fund the Department of Homeland Security after Democratic leaders  dropped an earlier pledge to block it unless they get assurances from House Republican leaders that it would pass their chamber.

The bill advanced on a procedural vote by a 98-2 margin. The only dissenters were Republican Sens. James Inhofe (Okla.) and Jeff Sessions (Ala.)

Final passage could come before Friday when current funding will run out at DHS.

"We're going to do everything we can to make sure it passes by an overwhelming vote," Senate Minority Leader Harry M. Reid (D-Nev.) told reporters.

He said he looked forward to working with Republicans "in the next 24 hours" to get this done. ​

Even if the bill clears the Senate,  it's unclear how the House will respond. Speaker John A. Boehner (R-Ohio) told reporters Wednesday morning that he was waiting to see what the Senate does before making his next move.

The House has passed a DHS funding bill that also takes aim at President Obama's executive actions on immigration. Senate Democrats have repeatedly blocked that bill.

After the Senate addresses DHS funding, Senate Majority Leader Mitch McConnell (R-Ky.) plans to hold a vote on a standalone measure that would undo Obama's 2014 actions, which included stemming the deportation of millions of undocumented immigrants.

But Reid said he would encourage Democrats to block that bill until a DHS funding bill passes both chambers of Congress.