WASHINGTON — The aim has become very simple for House Republicans stumbling closer to passing a bill to revise the Affordable Care Act: just get it off their plates and over to the Senate.

In the messy effort to rally their often unruly party around a measure to replace big parts of President Barack Obama’s health-care law, House leaders have been forced to leave other objectives by the wayside and focus on one simple, political goal: pass a bill they can say repeals Obamacare – even if it has no hope of survival in the Senate – to shield their members in next year’s elections.

Vice President Mike Pence applauds as President Donald Trump walks off after speaking to the Independent Community Bankers Association, Monday, May 1, 2017, in the Kennedy Garden of the White House in Washington. (AP) Associated Press/Evan Vucci

“I would hope it gets changed over there,” Rep. Peter T. King, R-N.Y., told Bloomberg News, echoing other center-right members who explicitly said they were willing to pass the new revision in hopes that the Senate would strip out the harsher provisions.

Even that goal, however, is proving elusive. By late Monday, House leaders had collected more votes than ever but still appeared to be shy of the 216 Republicans they need to pass the measure. They’re stuck between conservatives and moderates, both keenly aware of how they can be attacked on the issue next year.

“If you’re in the House, what you should be thinking now is that if it doesn’t survive, it all comes back to you,” said Sen. Roy Blunt, R-Mo. “I think what they should be focused on is getting the process moving and, frankly, passing the obligation over to the Senate.”

The White House, where aides have suggested a Wednesday vote is possible, continued to lobby members Monday even though no vote had been scheduled. Vice President Mike Pence hunkered in his office on the House side of the Capitol, with undecided and yes-voting members stopping by to talk.

Rep. Joe Barton, R-Texas, a conservative member of the whip team who had endorsed the previous version of the bill, told reporters that the votes were there to pass the new version. But several members from swing seats, including Rep. Mike Coffman (R-Colo.) and Rep. David Valadao, R-Calif., said outside Pence’s office that they remained undecided.

Even some members who won their seats partially on promises to repeal the ACA are blinking, citing changes to the proposed replacement that would allow insurers to charge higher premiums to patients with pre-existing conditions if their state got permission from the feds.

In 2010, Rep. Billy Long, R-Mo., campaigned for a safe Republican seat in Congress by pledging to fight “government-run health care.” Every two years, he won easy victories while telling voters he was “fighting to repeal Obamacare.”

On Monday, Long came out against the American Health Care Act with a few kind words about the law it was designed to replace. During unrelated votes Monday night, Long could be seen in a lengthy conversation with House Speaker Paul D. Ryan, R-Wis.

“I have always stated that one of the few good things about Obamacare is that people with pre-existing conditions would be covered,” Long said in a statement. “The MacArthur amendment strips away any guarantee that pre-existing conditions would be covered and affordable.”

Over the weekend, President Trump hadn’t helped. In an interview on CBS News’s “Face the Nation” Sunday, Trump said the latest bill would “beautifully” protect those with pre-existing medical conditions – which is not fully true.

As Republicans have struggled to find a health-care bill on which they can reach a consensus, Ryan agreed to support an amendment that would allow insurance providers in some states to deny coverage or charge higher premiums to people with pre-existing conditions or costly health problems, as long as that state set up “high-risk pools” that could help cover the cost of care.

Proponents have said this would lower premiums for healthy individuals, but critics have argued that it would dramatically drive up costs for those who are seriously ill. Proponents also noted that states can choose to leave current mandates in place.

Conspicuously absent from the House Republican effort to get to 216 is much talk about what happens in the Senate. There, Republicans will run up against the Senate parliamentarian, who must rule on whether some provisions are allowable in a budget reconciliation bill – the vehicle they’re using to repeal the health-care law to avoid a Senate rule requiring a 60-vote win that would require Democratic votes.

Senate Majority Leader Mitch McConnell, R-Ky., is likely to introduce a substitute version removing those provisions, just as he did back in 2015, when Congress passed a bill repealing the Affordable Care Act that President Barack Obama then vetoed.

“All of the policy considerations and policy constructs assembled by the House over the past couple of months may become moot,” said Chris Jacobs, who advised the House Republican Conference on health policy while the 2010 health-care law was being passed.

Sen. Charles E. Grassley, R-Iowa, said the House should move “quickly” on passing its bill. “I’m not going to tell them what to do, but I am going to say that if they don’t move pretty quickly, we oughta see what we can do in the United States Senate,” Grassley said.

But Sen. Rand Paul, R-Ky., the one Republican who opposed a Senate test vote on repeal in January – arguing that it did not go far enough – warned that even the new version of the AHCA fell short of his standard.

“It still could be improved a great deal, but it’s an open question of whether the Senate would fix it or make it worse,” said Paul. “I’m not excited about having taxpayer money going to insurance companies. That was a big part of Obamacare, and it’s a big part of this.”

There’s also no talk of getting a score from the Congressional Budget Office on how the changes would affect the cost of the bill or how many Americans it would cover, even though Republicans came under heavy fire in March for advancing their original measure without an estimate from Congress’s official scorekeeper.

Sen. Ted Cruz, R-Texas, dismissed concerns that the Republican bill will have to undergo major revisions in the Senate – or that Republicans won’t even be able to pass it in the House.

“Legislating takes time,” he said. “It’s worth remembering it took Obama 14 months to pass Obamacare,” he said. “The House repeal bill was on the floor for 14 days. That’s not nearly long enough to draft legislation as consequential as this.”

Despite the resistance from some members, House Republicans can’t get around the fact that for seven years, they have promised to repeal the Democrats’ health-care law. As the pressure mounts, they’re striving to just get the bill passed and let the Senate worry about how it could actually become law.

Sen. Bill Cassidy, R-La., noted that reality in a Monday interview with CNN. He said the House bill remains a “work in progress” and that some of it would be scrapped in negotiations.

“The House has to pass a bill,” said Cassidy, who along with Maine’s Sen. Susan Collins has written a replacement bill that retains much of the Affordable Care Act. “It’ll go to conference committee. I’m sure the administration will be involved. There will be two other times when what the White House is advocating can be addressed.”

Cassidy was skeptical of the modified AHCA, which creates high-risk pools for people with pre-existing conditions.

“I suspect the advocates for the bill will say that’s their guarantee,” he said. “I will insist that the president’s pledges be met. And the president pledged that he would take care of people with pre-existing conditions.”

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