(CNN) President Donald Trump touted progress as top House Republicans may be nearing a significant breakthrough on efforts to repeal and replace Obamacare, one month after a Republican health care bill was pulled from the House floor.

The leader of the conservative House Freedom Caucus, North Carolina Rep. Mark Meadows, and the head of the moderate Tuesday Group, New Jersey Rep. Tom MacArthur, are working toward a deal that could bring 18 to 20 new "Yes" votes from the conference's conservative wing, according to a source familiar with the talks. But it's not clear there would be enough votes in the broader GOP House conference to pass the bill.

The White House and GOP leadership have been involved in the talks and are aware of the latest progress, the source added.

A White House official confirmed to CNN the administration is aiming to circulate legislative text on a health care replacement plan by Friday or Saturday, though exact timing is still fluid.

During a news conference Thursday, Trump said "we're doing very well on health care," that the legislative effort is "evolving" and that "there was never a give-up."

"I want to get both. Are you shocked?," he said in reference to health care and passing a continuing resolution to fund the government next week.

There will be a House GOP conference call on Saturday to discuss the upcoming legislative agenda following the Easter recess, according to a Republican lawmaker, where health care is expected to come up.

GOP Rep. Tom Cole of Oklahoma, who supported the Republican health care legislation known as the American Health Care Act, has been in touch with his colleagues in the House as well as the White House about health care over Congress' two-week recess, which wraps next week. His big-picture takeaway on the discussions: "Differences have narrowed and this thing is very much alive."

But in a phone interview with CNN Thursday morning, Cole also cautioned that there are a lot of mixed feelings across the conference among members and aides on whether this effort could be any more successful than the first time. Much of the concern continues to center on gutting Obamacare's protections for those with pre-existing conditions.

"It's amazing the diverse things you hear from different people," Cole said, noting the responses range from "'it's hopeless' to 'Oh, it's going to get there.'"

The colleagues he has spoken with appear "cautiously optimistic," Cole added. "Some of the folks in leadership -- they've been burned so many times they're afraid to be optimistic," he said.

As of now, the broader conference is not aware of what the new health care blueprint is, suggesting that things are very much in the air. "I think only the people directly involved in the talks have a very clear idea of what's happening," Cole said.

A Republican member familiar with the negotiations expressed some skepticism that the Freedom Caucus can actually deliver the votes they are promising.

"They have yet another agreement in principle, but no final legislative language," the member told CNN. "We have no idea if this nets us votes or can pass the Byrd Rule," a reference to whether the current bill would be eligible to pass via reconciliation, which would only need a simple majority to pass in the Senate, as oppose to needing to clear 60-vote threshold.

The member added, "I would say we are skeptical at best."

In a speech in London on Wednesday, House Speaker Paul Ryan said that "health care is not dead."

Ryan said he and others are working on "finishing touches" to their repeal and replace legislation, though he also acknowledged that this work is "difficult."

In the past few weeks, the White House has been making a strong push to take one more stab on health care, particularly as Trump approaches his 100-day mark in office.

But there are significant obstacles.

The focus next week will be to passing a bill to avoid a government shutdown, so it is unclear how much members will be able to focus on getting another health care vote in the House.

And there are serious philosophical divides between the House's more moderate and conservative members.

Meadows and his colleagues on the Freedom Caucus want to get rid of as many of Obamacare's insurance reforms, known as Title One for their section in the Affordable Care Act, as possible. These measures contain the main protections for those with pre-existing conditions, as well as more general consumer protections that ensure enrollees get comprehensive benefits and limit their financial liability if they get sick.

The original bill ran aground last month after Freedom Caucus members pushed at the last minute to scale back or eliminate these protections, which they argue is the only way to lower premiums. But moderates, including the Tuesday Group, balked, saying that this could severely harm the care of those with pre-existing conditions -- one of the most popular parts of Obamacare.

The talks between Meadows and MacArthur have continued to focus on three areas identified before the congressional recess: essential health benefits , which requires insurers to provide 10 services, including maternity, substance abuse and prescription drugs, in all plans; community rating, which bans insurers from charging more to people based on health history or gender, and guaranteed issue, which mandates insurers cover everyone regardless of their medical status, according to sources.

Lawmakers are looking at allowing states to obtain waivers to opt out of some of these protections, but requiring states to attest that the waiver's goal is to reduce the cost of health care or increase the number of insured. Under the so-called MacArthur Amendment, states could never allow insurers to price plans based on gender or to turn away people because of pre-existing conditions.

Members are also discussing providing extra money to help defer states and insurers' costs for caring for the sick through high-risk pools or other mechanisms. The current deal-making has centered on what triggers would allow states to opt out and how much additional funding would be provided. Any states that allow insurers to charge consumers more based on health status would also have to set up a high-risk pool to assist those affected.

In addition to concerns about those with pre-existing conditions, some moderate Republicans were also upset that the House bill would severely cut back funding for Medicaid, which provides health coverage for many opioid addicts. Several GOP governors in states that expanded Medicaid to low-income adults were pushing representatives to revise these provisions in the House bill.

The Congressional Budget Office estimated that 24 million fewer people would be covered under an early version of the Republican bill, which did not include all these changes to insurance reforms. Much of the coverage loss stemmed from the reductions to Medicaid.

Americans have not embraced the House bill that was pulled last month. Only 17% of voters approved of the plan, according to a Quinnipiac poll released last month

A new Quinnipiac poll found that only 36% of American voters say Republicans in Congress should try again to repeal and replace Obamacare, while 60% say the Republicans should "move on."

Voters disapprove by a 65% to 29% margin of the way Trump is handling health care.

The March Quinnipiac poll was conducted March 16-21, and consisted of 1,056 respondents and had a margin of error of plus or minus 3 percentage points. The new poll was conducted April 12-18 among 1,062 respondents in the US and has a margin of error of plus or minus 3 percentage points.