The White House told congressional leaders Wednesday that the administration will continue making Obamacare cost-sharing payments, eliminating the biggest sticking point as lawmakers rush to pass a final 2017 spending bill and avert a government shutdown looming at the end of the week.

Democrats had wanted to include the funds in the spending bill, but Republicans had balked, saying they’d never been included before — though President Barack Obama had doled out the money anyway, in what a judge has ruled an illegal expense.

The Trump administration’s promise to continue the payments without congressional approval preserves a GOP anti-Obamacare lawsuit, but still keeps the money flowing, which Democrats said was their most important goal.

“Our major concerns in these negotiations have been about funding for the wall and uncertainty about the [cost-sharing] payments crucial to the stability of the marketplaces under the Affordable Care Act. We’ve now made progress on both of these fronts,” House Minority Leader Nancy Pelosi said.

Mr. Trump agreed earlier this week to pull his demand that money for his wall be included in the spending bill. That put pressure on Democrats to drop their insistence on Obamacare funding.

But Mrs. Pelosi said a deal is still proving elusive.

“More progress needs to be made on some of our priorities, and we continue to be concerned about poison-pill riders that are still in this legislation,” she said. “Our appropriators are working in good faith toward a bipartisan proposal to keep government open.”

The government has been running on stopgap funding since Oct. 1, or nearly seven months. Congress needs to pass a new bill by midnight Friday or else the government will face a partial shutdown.

Late Wednesday, House Republicans announced a one-week stopgap bill that would carry the government through May 5. That bill, which needs to pass by Friday, would buy more room for negotiations.

Obamacare payments emerged as a stumbling block earlier this month after Mr. Trump threatened to withhold the money as a negotiating tactic on the broader health care repeal. Democrats demanded certainty in the money, saying the health markets count on those payments to make Obamacare work.

House Republicans, however, had challenged the payments as illegal, saying Congress had specifically nixed the money. A federal judge agreed with them, ruling Mr. Obama acted illegally.

Including the payments in the new spending bill could have invalidated that lawsuit — though even some Republicans said it wasn’t fair to end the payments now while the rest of Obamacare is intact.

“A lot of us, myself included, think those subsidies ought to be continued until we have an alternative. I think we owe that to the companies, I think you owe that to the people in the health care system,” said Rep. Tom Cole, Oklahoma Republican.

Having the White House agree to make the payments was the middle-ground solution.

Senate Minority Leader Charles E. Schumer said Wednesday morning that there were still other stumbling blocks, including federal support for a health fund for retired miners and money to bolster Puerto Rico’s struggling Medicaid system.

An even bigger hurdle, though, appears to be the personalities involved.

Majority Leader Mitch McConnell said Tuesday he’d been pushing Mr. Schumer to negotiate directly with the White House to try to get a bill done, but said the Democratic leader was refusing even to talk to the president.

Mr. Schumer countered that was because he wasn’t going to negotiate at all as long as Mr. Trump was seeking funding for the border wall.

“Senator McConnell wanted me to negotiate the wall with President Trump. I said to him two things: First, it’s not a negotiation. No wall. And second, I said, only you can persuade him that he shouldn’t do the wall because it’ll cause a government shutdown,” Mr. Schumer said.

And Mr. Trump himself took to Twitter on Wednesday evening to blame the Democrats’ demands for the difficult negotiations.

“Democrats are trying to bail out insurance companies from disastrous #ObamaCare, and Puerto Rico with your tax dollars. Sad!” he wrote.

Democrats’ disdain for Mr. Trump bled over into the House as well, where Mrs. Pelosi met with White House Budget Director Mick Mulvaney on Tuesday night, and then attacked him as an obstructionist a day later.

“Congressman Mick Mulvaney was a chief architect of the government shutdown in 2013. He not only voted to shut down government but voted against ending the shutdown. He also voted to default on the full faith and credit of the United States,” Mrs. Pelosi said.

Democrats are confident that if there is a government shutdown, the GOP would take the blame, emboldening the minority party to press its demands.

But Sen. Ted Cruz, Texas Republican, said Democrats are the ones angling for a shutdown, believing it will energize their base.

“I think Chuck Schumer and the Democrats want a shutdown,” he continued, “I think they’re trying to provoke a fight.”

• Tom Howell Jr. and Sally Persons contributed to this article.

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