"It’s pretty much everything I was looking for in terms of concessions," said Representative Scott DesJarlais of Tennessee, a member of the conservative House Freedom Caucus who had opposed an earlier version.

House Speaker Paul Ryan of Wisconsin and his lieutenants don’t know whether the new converts will add up to enough support to pass the bill.

"We’re showing people the language now," Majority Leader Kevin McCarthy of California said Tuesday. "And as we work through and talk through, we’re going to talk to more members tonight and tomorrow. Check with me tomorrow and we’ll be able to know."

Republicans haven’t yet conducted an updated vote count, according to several House GOP members who discussed the measure in a closed-door meeting Tuesday.

Still ‘cautious’

"Cautious," said Representative Phil Roe of Tennessee, a medical doctor, of the approach that House Republican leaders are taking. Representative Steve Chabot of Ohio said everyone is proceeding quietly so that nothing happens "to blow everything up."

Related: White house still pressing to hold ACA vote this week

But an influential Republican moderate, Charlie Dent of Pennsylvania, said the changes don’t soften his opposition to the GOP bill.

"The amendment as I understand it doesn’t change my position. I am still a no," he said Tuesday.

Changes to the bill may also make it more difficult to pass the Senate. “It will be harder for the Senate to get 51 Republicans,” Republican Senator Roy Blunt of Missouri, a former House majority leader and longtime whip, said Wednesday

The White House, which has been involved in discussions about the changes, is still eager to resurrect the health-care bill.

“We’re not going to overpromise anything; when the votes are there, the speaker will bring it to the floor but no sooner than that," White House Chief of Staff Reince Priebus told reporters late Tuesday. He said he didn’t know if that might be this week or next.

Repeal of the Affordable Care Act was a major Trump campaign promise and a longtime goal of House Republicans.

Growing support

"We probably had about half of the members of the Freedom Caucus in the first go-around," White House legislative affair director Marc Short told reporters Tuesday. "With this amendment, I’d like to think we have greater than 80 percent -- we are very confident in that."

Short said he still thinks they could get the health-care bill passed before the GOP tax bill is introduced in the next four to six weeks.

Much of the renewed optimism stems from new support within the House Freedom Caucus, a group of about three dozen staunch conservatives.

Opposition from many inside that group, who wanted a more robust repeal measure, along with skepticism from many Republican moderates, was pivotal to Ryan’s decision last month to abruptly scrap a vote on the bill for lack of votes.

On Tuesday, Freedom Caucus’ chairman, Mark Meadows of North Carolina, had a meeting at the White House, and told reporters afterward, "There is a lot of optimism."

Other members of the Freedom Caucus who previously didn’t support the bill, including Representative Dave Brat of Virginia, and DesJarlais, said they now support the measure.

Both pointed to changes hashed out over a two-week recess by Meadows and Representative Tom MacArthur, a New Jersey Republican and co-chairman of a group of House moderates.

Their amendment was being circulated among members Tuesday and is expected to be discussed at a Wednesday morning GOP conference meeting and a Freedom Caucus gathering later in the day.

State waivers

The amendment would allow insurers to charge higher premiums to people with pre-existing conditions in states that get a waiver. To obtain the waiver, states would have to provide sick people priced out of commercial insurance access to a so-called high-risk pool run by the federal government, or establish their own, and satisfy other conditions.

Related: Trump changing tactics on tax after health care repeal failure

Said Brat of his possible support: "If it shows up in language the way we discussed it, yeah." A founding Freedom Caucus member, Jim Jordan of Ohio, said the group as a whole will hold its own meeting on Wednesday about the bill.

Even Mo Brooks of Alabama, a conservative who opposed the previous version, said he’s considering the amendment.

"I believe there will be some movement," he said. "I don’t know how much."

But it’s far from clear if the Freedom Caucus will take a united position backing the amended bill. And anything short of united support could cause problems, given the continued opposition of a number of moderates who don’t see the change as an improvement and continue to have concerns about the Medicaid cuts and other issues central to the bill.

Dent said the bill doesn’t provide a "soft enough" landing for states that expanded Medicaid, and still doesn’t provide sufficient support to help low- and middle-income people, in his view.

Pressure from the White House, combined with fresh support from conservatives, could put intense pressure on moderates to vote for the bill. But those moderates in swing districts, unlike the members of the Freedom Caucus, could end up losing their seats if the repeal bill continues to remain unpopular.

"I will vote my conscience," said Representative Leonard Lance of New Jersey, who said he also remains opposed.

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