“There's … general agreement that those who are brought into legal status will not be brought in subject to further dollar commitments as a result,” Hoyer, the Democratic whip, told reporters in the Capitol.

Hoyer said the House group, which last week announced an “agreement in principle,” is still ironing out significant details, including how to approach the health benefits question.

"”They're still talking about this; they're still discussing [it],” Hoyer said. “When you have an agreement 'in principle,' obviously that means you have an agreement on sort of a broad brush. But that needs to be implemented by specifics, and I think there's still discussion about those specifics.”

The comments are just the latest indication that the House group might be further from finalizing their package than negotiators have suggested in recent days.

Last week, for instance, Rep. Mario Diaz-Balart (R-Fla.), one of the eight House negotiators, said the process was complete except for the final drafting of legislative language.

“Since we're done, it's just an issue of getting the bill back and rereading it,” he said. “There's no disagreement at all. … There's nothing that somebody's going to want to put in or change that it's going to blow this thing up because it's stuff that is, frankly, … disposable. … Nothing that would kill it.”

But the pushback from Democratic leaders against the GOP's healthcare provisions could prove significant. Although the Republicans control the lower chamber, conservative members have been wary of an immigration overhaul, and it remains unclear if GOP leaders could pass such a package without significant Democratic support.

Rep. Xavier BecerraXavier BecerraOvernight Energy: New controversies cap rough week for Pruitt | Trump 'not happy about certain things' with Pruitt | EPA backtracks on suspending pesticide rule EPA backpedals on suspending pesticide rule following lawsuit Overnight Health Care —Sponsored by PCMA — Spotlight on Trump drug pricing plan MORE (Calif.), the only Democratic leader who's a member of the negotiating team, has also been the only member of the group reluctant to say a deal is done, even in principle. Becerra on Tuesday remained reticent on that issue, though he suggested any remaining policy disputes are minor.

“It's not so much that there are any concerns, it' just the process. We're trying to make sure that we can get to a bill that we can put on the floor,” Becerra said. “We're on the precipice of doing something really good.”

Hoyer, for one, remains optimistic that a deal can be reached.

“We are for a comprehensive immigration bill, and I'm hopeful that we can come to an agreement on the specifics,” he said.