(CNN) The Trump administration, under pressure from conservative lawmakers, appears increasingly open to changing legislation repealing and replacing Obamacare in order to placate the right -- but that may guarantee at least this version of the bill is as good as dead in the Senate.

The political calculation is setting off a domino effect on Capitol Hill.

Just days to go before a Thursday vote on the House GOP bill, moderate Republicans have already expressed deep reservations about making drastic reforms to the current health care system. Making additional changes to satisfy conservatives is certain to strengthen opposition among centrist Republicans.

But moving the bill to the right dooms the proposal in the Senate, where the GOP has a razor-thin majority.

For now, the White House appears fixated on shepherding the bill repealing the Affordable Care Act through the House, with a vote on the seven-year anniversary of President Barack Obama signing the law. In its 11th hour negotiations, Trump and his top deputies have made compromises to appease the GOP's most conservative faction.

After meeting with members of the conservative Republican Study Committee this week, Trump announced that the lawmakers in the room who had previously been "no" votes were now on board, thanks to legislative changes that the White House and House GOP leaders had agreed to.

The negotiations continued over the weekend.

Several conservative lawmakers opposed to the House GOP health care bill -- Rep. Mark Meadows, chairman of the House Freedom Caucus, and Sens. Ted Cruz and Mike Lee -- met with senior White House aides on Saturday at Mar-a-Lago.

Trump himself did not participate in the meeting, a White House official said. But the President's senior most aides -- chief of staff Reince Priebus, chief strategist Steve Bannon and policy aide Andrew Bremberg -- were present, according to a senior Republican familiar with the meeting.

'Bannon is definitely our ally'

Cruz, Lee and Meadows made the case for going further with the American Health Care Act, the name of the bill introduced by House Speaker Paul Ryan and other senior Republicans, and backed by Trump. According to the Republican source, discussion included phasing out Medicaid expansion earlier and repealing Obamacare rules like essential health benefits and requiring insurers to allow 26-year-olds to stay on his or her parents' insurance -- one of the most popular provisions of the Affordable Care Act.

Notably, Bannon appeared receptive to the conservative contingent, while Priebus and Bremberg advocated for Ryan's approach.

"Bannon is definitely our ally in the White House," the Republican source said.

Health and Human Services Secretary Tom Price said on ABC about the negotiations in the House: "It's a fine needle that needs to be thread. There's no doubt about it."

It's not clear whether the White House will announce further concessions following the Saturday gathering in Florida. A Freedom Caucus source simply told CNN on Sunday that the negotiations are continuing "in good faith."

But giving more ground to conservatives risks a hemorrhaging of support from House Republican moderates, who are already uncomfortable with and have said they cannot support the legislation in its current form.

Particularly troubling to more centrist Republicans is the Congressional Budget Office's prediction last week that the House legislation would lead to some 24 million more people being uninsured by 2026 than under Obamacare, as well as the projection that premiums would spike for vulnerable demographic groups like older and low-income Americans.

DOA in the Senate?

A more conservative version of the bill that Ryan originally unveiled would most certainly be dead on arrival in the Senate.

Multiple Senate Republicans have publicly aired grave concerns about restructuring Medicaid. Under Obamacare, 31 states opted to expand the program for low-income adults and 16 of those states have Republican governors.

GOP Sen. Susan Collins of Maine, for example, said this week that she could not support the House bill in its current form, particularly because of its effects on the ability of low-income seniors to afford health insurance.

The White House, too, has admitted that the House bill would have trouble in the Senate. Trump administration officials acknowledged to Republican senators at a White House meeting earlier this week that "they don't have the (Senate) votes to pass this in current form."

In an interview on CBS' "Face the Nation" on Sunday, Cruz briefly discussed his Mar-a-Lago visit, saying the meeting with the President's team spanned three hours. He added that he had spoken multiple times with Trump and Vice President Mike Pence about the bill.

"I am spending night and day meeting with House members, meeting with senators, meetings with the administration," Cruz said.

Cruz and others have advocated publicly for a much more sweeping, conservative bill, but said he had not managed to sway Trump yet. He described Trump as willing to listen to everyone and that the bill was still in an ongoing negotiation process.

"I think that the President right now is listening to the arguments on all sides," Cruz said.