The lawmakers spoke after last week's stunning crash of the GOP's drive to tear down President Barack Obama's 2010 health care law and replace it with their vision of more limited federal programs.
While the leaders stopped short of saying they were surrendering on an issue that's guided the party for seven years, their remarks underscored that Republicans have hit a wall when it comes to resolving internal battles over what their stand should be.
No. 2 Senate GOP leader John Cornyn of Texas signaled displeasure at White House budget chief Mick Mulvaney, who pushed senators in weekend TV appearances to keep voting on health care until they succeed.
Mulvaney has "got a big job, he ought to do that job and let us do our jobs," Cornyn said. He also said of the former House member, "I don't think he's got much experience in the Senate, as I recall."
Despite Mulvaney's prodding and weekend tweets by President Donald Trump insisting senators revisit the issue, even the White House's focus turned Monday to a new horizon: revamping the tax code.
White House legislative director Marc Short set an October goal for House passage of a tax overhaul that the Senate could approve the next month. Plans envision Trump barnstorming the country to rally support for the tax drive, supported by conservative activists and business groups heaping pressure on Congress to act.
In the House, 43 Democratic and Republican moderates proposed a plan that includes continuing federal payments that help insurers contain expenses for lower-earning customers. It would also limit Obamacare's requirement that employers offer coverage to workers to companies with at least 500 workers, not just 50.
Trump has threatened anew in recent days to cut off the payments to insurers, which total $7 billion this year and are helping trim out-of-pocket costs for 7 million people.
Those payments to insurers have some bipartisan support because many experts say failing to continue them or even the threat of doing so is prompting insurers to raise prices and abandon some markets.
Obama's statute requires that insurers reduce those costs for low-earning customers.
Kristine Grow, spokeswoman for the insurance industry group America's Health Insurance Plans, said Monday that halting the federal payments would boost premiums for people buying individual policies by 20 percent.
Hoping to find some way forward, health secretary Tom Price met with governors and Louisiana Republican Sen. Bill Cassidy. Among those attending was Republican Arizona Gov. Doug Ducey, who's been trying to defend his state's expansion of Medicaid, the health insurance program for poor people, against proposed GOP cuts.
Cassidy and Sens. Lindsey Graham, R-S.C., and Dean Heller, R-Nev., have proposed converting the $110 billion they estimate Obama's law spends yearly for health insurance into broad grants to states.
Police brutality? White House says Trump was 'making a joke'
Washington • The White House defended President Donald Trump's recent remarks that police shouldn't be too nice when transporting suspects, saying Monday that the president was "making a joke."
On a visit to Long Island, N.Y., last week, Trump implored police officers, "Please don't be too nice." He said some officers are too courteous to suspected criminals when arresting them.
"Like when you guys put somebody in the car, and you're protecting their head, you know, the way you put your hand over" their head, he said, putting his hand above his head for emphasis. "I said, 'You can take the hand away, OK?"
His remarks prompted critics to accuse the president of encouraging police brutality.
White House press secretary Sarah Huckabee Sanders told reporters Monday, "I believe he was making a joke at the time."
Trump touts himself as a president who is "strong on law enforcement" and tough on crime.
The Associated Press