Senate Democratic leaders met face-to-face with every one of their members this week, Majority Whip Dick Durbin (D-Ill.) said Sunday.

Durbin, Democrats' top vote-counter in the Senate, said that he and other party leaders had met with each one of their senators on healthcare to ensure that there was a strong majority to move forward with legislation the House is expected to pass Sunday.

"This last week, Harry Reid, our leader, Sen. Schumer and Sen. Murray and I have met with in fact every Democratic senator face-to-face so that we understand the importance of this vote," Durbin said during an appearance on CBS's "Face the Nation."

Senate Majority Leader Harry Reid (D-Nev.) released a letter to House Democrats pledging to follow through with a vote on a series of changes to the original Senate healthcare bill. Reid said that a "significant majority" of senators had signed onto those changes.

"Let me tell you something: I think there may be even more than 51," Durbin said Sunday.

The House will vote to pass the Senate's healthcare bill on Sunday, as well as a series of changes to it under the budget reconciliation rules. Under those rules, the Senate could pass that measure with only a simple majority of votes, instead of the 60 votes normally needed to end a filibuster.

Durbin said he expected the Senate to begin working on that new healthcare effort this week, though he said he couldn't predict how long that would take, citing the bevy of amendments and points of order Republicans have said they plan to raise against the legislation.

"I can't say what the parliamentary rulings will be on every position, but I can say that we have a majority of the United States Senate ready to vote for the reconciliation bill to move this forward for the president's signature," Durbin said. "If there's some procedural problem we run into, we're going to work through it."