Clyburn, D-S.C., acknowledged it would be a tough road to passage, as he and other party leaders try to win over House members who haven't committed to voting in favor of the Senate bill.
"We don't have (the votes) as of this morning, but we've been working this thing all weekend. We'll be working it going into the week," Clyburn said on NBC's "Meet the Press."
"I have been talking with members for a long time on this, and they have the will do it. They've been looking to us to create a way to do it. We've gotten to a place where we do have a way to do it, and I think members are going to vote for this."
Some House Democrats worry the Senate won't approve changes to the bill after the House approves it.
But Senate Majority Whip Dick Durbin, D-Ill., who also appeared on the program, called those concerns unjustified.
"We are in the process of contacting every single Democratic senator," Durbin said, adding that when House Speaker Nancy Pelosi, D-Calif., goes before her caucus, "it will be with a solid assurance that when reconciliation comes over to the Senate side we are going to pass it."
Notable deaths of 2014 [PHOTOS]
N.J. man whose son shot playmate pleads guilty