Washington (CNN) Conservative House members say they got a promise from leadership to pursue a separate hard-line Republican-only immigration bill in exchange for their votes to pass government funding Thursday night -- a measure that several Republicans doubt could pass the House, let alone the Senate.

The bill is a proposal from key committee and subcommittee chairs Bob Goodlatte, Raul Labrador, Mike McCaul and Martha McSally that includes a large number of hard-line immigration provisions that Democrats and some Republicans have said are nonstarters.

Labrador told CNN after votes Thursday that the commitment to members of the House Freedom Caucus was "to actually whip (the bill) and to work on it like they did on all the other important bills we've done this year." While there was no clear timeline for a floor vote, Labrador said that "should" be the goal.

"if people want to vote against it, they can explain to their constituents why they're voting against a bill like this," Labrador said.

The immigration bill includes a number of controversial pieces, including mandatory worker verification, cracking down on sanctuary cities, changing asylum thresholds and cutting legal immigration to the US by 25%. The bill offers Deferred Action for Childhood Arrivals recipients indefinitely renewable three-year work permits but no pathway to citizenship.

Read More