Pence and two top White House officials made the offer Monday night in a closed-door meeting with members of the House Freedom Caucus, participants said.
The Freedom Caucus is comprised of almost three dozen members, many of whom opposed the AHCA, developed by House Republican leaders including Speaker Paul Ryan (R-Wis.), on the grounds that it was simply another entitlement program, nicknaming the bill "Obamacare Lite". In the meantime, caucus chairman Mark Meadows, R-N.C., called Monday's session a "good meeting".
"But that is not to say that we are ready to go".
A Republican senator playing a major role in the attempt to revive the healthcare bill, Rand Paul, said on Tuesday that the Trump administration and Republican lawmakers were still at an "impasse" in talks. "While a comprehensive approach is preferable, some of the problems can be fixed with more narrow reforms, and in a bipartisan way", the bishops said, suggesting that Congress pass the Conscience Protection Act, extend full Hyde Amendment protections to the Affordable Care Act, and enact other targeted laws to remove current and impending barriers to obtaining health care.
Rep. Gosar said the Trump-Ryan plan still had too many "mandates and regulations" that have increased the cost of insurance.
Over the past seven years, for example, opponents of Obamacare have made much of the fact that the Affordable Care Act was passed without any Republican support.
While Freedom Caucus members say they are encouraged by the offer, none of them have committed support, saying they want to wait to review the legislative text first.
While the White House has downplayed the significance of the negotiations, a Freedom Party member told the Washington Examiner that he was "cautiously optimistic that we may be able to reach a path forward by week's end".
Republican lawmakers said the new push on healthcare would maintain Obamacare's essential health benefits clause listing services and care that insurers must cover.More news: Damaging winds, hail, weak tornado possible tonight
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It also remains to be seen how this change, meant to win over conservative critics of the pulled repeal plan, will be seen by moderate Republican lawmakers who opposed that plan for being too harsh.
Trump aides also discussed directing funds from the $115 billion stability fund for states into high-risk pools for people with pre-existing health conditions to better ensure insurance premiums come down in cost, Collins said.
And those 20 million people the Democrats said would lose health care coverage under the Republican bill will indeed lose their insurance.
Also at the evening meeting with conservatives were White House chief of staff Reince Priebus and budget director Mick Mulvaney.
Trump has lashed out at members of the conservative Freedom Caucus who contributed to the defeat of the bill.
"I would like to get this finished", Walters told The Hill. But this move spooked moderate Republicans, who were concerned it would leave their constituents with less care and bigger bills, and led to the bill being shelved. After dealing initially dealing with the House Freedom Caucus with a carrot, Trump has settled on a stick, promising to "fight" Freedom Caucus members along with Democrats in 2018.
"For every adjustment you make one way, it has an equal and opposite reaction the other way", he said.
"I think it's important for the administration to signal early the general shape" of what they would like to accomplish so that there are fewer proposals vying for attention, said Michael Mundaca, a former assistant Treasury secretary now at Ernst & Young. Ryan, in a news conference Tuesday morning, said "it's all about getting premiums down".
"We are not now negotiating with the Freedom Caucus".