“Obviously that’s a direction I think that a lot of our members want to move, to keep some of those in place and use the revenues to put into other places in the bill where it can make a difference,” John Thune of South Dakota, the No. 3 Senate Republican, said Tuesday. The decision to explore changes was first reported by the Wall Street Journal.

The reversal on tax cuts, aimed at winning over moderate holdouts, comes amid an announcement Tuesday by Senate Majority Leader Mitch McConnell that he is delaying a planned summer recess by two weeks, giving Republicans more time to pass a health care bill.

McConnell told reporters that he plans to release his revised bill on Thursday, with a new CBO estimate and an important procedural vote coming next week.

But Republican Lindsey Graham of South Carolina said Tuesday that he plans to release contours of an alternative health bill this week and that he is seeking the support of governors and senators of both parties.

“I want to do the best I can, and I think the best we can is not on the table right now,” Graham said.

A number of moderate Republicans had recoiled from the bill after the CBO estimated that 22 million fewer people would have insurance in a decade, and that premiums and deductibles for millions of low-income people would soar.

Related: Senate delays August recess amid unfinished health care plan

Closed-door talks

McConnell has been negotiating with his Republican colleagues over revisions after more than a half-dozen of them objected to a plan last month that combines tax cuts with deep reductions in health spending.

Although Republican leaders decided to retain two of the Obamacare taxes, others would still be repealed in their revised bill.

"The taxes that will be repealed are all the taxes that have been driving up the cost of insurance," said Senator John Barrasso of Wyoming, the No. 4 Senate Republican.

Other changes under consideration include revising Medicaid cuts and adding more spending to stabilize premium costs in the individual insurance market, according to a GOP aide who requested anonymity.

Republicans are waiting to hear back from the CBO before deciding whether to include an amendment proposed by Republican Senator Ted Cruz of Texas.

Cruz wants to allow insurers to offer cheap, bare-bones plans alongside those that meet the more comprehensive coverage requirements of Obamacare. Critics in both parties say the proposal would essentially put people with pre-existing conditions in the Obamacare insurance pool and allow young, healthy people to buy cheaper plans in a separate pool.

“It’s important to a number of our conservative members who want to see that in a final product,” Thune said. “We have other members who have a different point of view.” He added, “It’s something we want to explore” to potentially give consumers more choices.

Republican leaders have said they want to hold a vote before the recess, which has now been pushed into mid-August, and are prepared to move on to other issues, including a tax-code rewrite, if they can’t develop consensus around a proposed replacement to the 2010 Affordable Care Act.

Related: Poll: Americans don't want to lose emergency services, other EHBs

Asked if GOP leadership will move forward with the vote even if they lack the 50 senators needed to secure passage of the legislation, Barrasso said, "I’m convinced that there’ll be a vote next week to move to get on the bill."

If there were an imminent vote in the Senate, House Republican leadership would keep their chamber in session into the beginning of August, according to a GOP aide familiar with the plans. However, if the Senate works further into August and manages to pass a health care bill, there would be intense pressure from the Trump administration to bring House members back to Washington for another vote, according to the aide.

For the bill to pass, and Republicans to live up to their promise to eliminate President Barack Obama’s signature domestic accomplishment, they can lose no more than two GOP votes from their 52-48 majority amid unanimous Democratic opposition.

Skeptical Republicans

A number of Republicans have been pessimistic about the prospects in recent days.

Republican Senator Susan Collins of Maine, who opposes McConnell’s earlier bill, said in an interview Monday that she was heartened at the majority leader’s suggestion last week that Republicans will have to work with Democrats on a scaled-back measure shoring up Obamacare’s insurance exchanges if the GOP bill dies. She said she’s concerned about the impact of proposed cuts to Medicaid under the broader plan.

"I believe we should not repeat the mistake that President Obama made in passing major legislation with no support from the other party," Collins said.

Senator Bill Cassidy of Louisiana said on “Fox News Sunday” that McConnell’s original plan is dead and that what happens with the rewritten version remains to be seen. He put the odds of passing a bill in July at “50-50.”

While moderates are balking at deep cuts to Medicaid and to subsidies for individual market consumers, conservatives are seeking looser regulations for such policies that are sold on Obamacare’s exchanges.

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