Bipartisan House negotiators emerged from a critical meeting Thursday signaling that they again have come to a tentative agreement on a comprehensive overhaul of the nation’s immigration system.

Coming just a week after a tentative agreement was nearly derailed over health care disagreements, however, the path ahead for the legislation is anything but assured. Members of both parties noted that although a major hurdle has been averted, further issues could arise as the legislation is drafted.

Rep. Raúl R. Labrador, R-Idaho, said the group of eight negotiators has resolved the contentious issue of how and whether immigrants waiting to become citizens should be able to obtain health insurance.

He said provisional citizens will not be eligible for Medicaid or taxpayer subsidies to join health insurance exchanges, but that in the event of an emergency, they can receive medical care without the threat of deportation as long as they pay the bill. He also said they will be mandated to provide their own insurance on penalty of deportation.

“The law already requires that in an emergency situation no one can be denied health care. That’s what I thought our agreement was two weeks ago, but apparently some people needed clarification on that,” Labrador said, exiting the meeting. “I don’t think much changed today. I think there was just some confusion about some details. But I think we’re all good.”

Other members were mum on details, but Democrats signaled that they, too, believed an agreement had been struck. “We agreed not to make detailed comments, but I’m smiling, not frowning,” said Rep. Zoe Lofgren, D-Calif.

Labrador’s characterization of the health care deal comports with what Minority Leader Nancy Pelosi, D-Calif., said earlier Thursday before the group met. She told reporters she is hopeful a bill can move to conference with the Senate before the August recess.

“We have said since day one — we said it in the Affordable Care Act and we’ll say it here again — that undocumented people will not have access to subsidies in the Affordable Care Act,” she said. “Secondly: no access to Medicaid. No cost to the taxpayer, that has always been the Democratic position. So any thought that we want to do something different than that is simply not true.”

Pelosi signaled that the overall talks are far from over. She raised an issue with E-Verify, an Internet program that would allow employers to substantiate an employee’s eligibility to work in the United States.

She noted that tentative language would revert provisional immigrants back to undocumented status if the program is not fully implemented on time.

“They have a trigger in there that says that … if E-Verify is not effectively accomplished in five years, then all of these people revert to the status they have now. I think that’s pretty drastic,” she said. “It’s not the immigrant’s responsibility to make sure E-Verify works.”

It is unclear whether this or other issues could stymie the process, as staff begins the arduous task of drafting legislation over the Memorial Day recess.

For his part, Labrador said he believes all outstanding issues have been resolved and that the eight negotiators are on board. He did, however, note that as language is drafted other issues could arise, and he pulled no punches when noting where he thinks the difficulties are originating.

“I think the people in there are working in good faith, and I hope they continue to do so. If leadership wants to blow it up, that’s up to their leadership,” he said.

Speaker John A. Boehner of Ohio made a similar claim at his weekly news conference earlier Thursday, noting that “there are people on both sides of the aisle who have done their best to try to undermine their ability to get to an agreement.”

But he said he supports the House group’s effort and a path to conference.

“The House is going to work its will on immigration. We’re not going to be stampeded by the White House or stampeded by the president,” he said. “I’m confident that we’ll have a solid work product that we can go to conference with the Senate.”

Senate Majority Leader Harry Reid, D-Nev., said Thursday he was consulting with Senate Judiciary Chairman Patrick J. Leahy, D-Vt., and members of the bipartisan immigration group to decide whether to bring the Senate bill (S 744) to the floor when the chamber returns from recess and continues work on the farm bill.

House-drafted legislation still must advance through the House Judiciary Committee, and Labrador, who sits on the panel, said he thinks it could do so without obstruction.

Judiciary Chairman Robert W. Goodlatte, R-Va., also signaled his support on Thursday, although he has said in the past that he prefers a piecemeal approach to an immigration rewrite.

“We’re going to move ahead no matter what, but we hope they do, we’re wishing them well in reaching an agreement,” he said.

Labrador said he expects legislation to materialize in the first few weeks of June, when Reid said he wants that chamber’s immigration bill to get a vote.