WASHINGTON — The abrupt decision Tuesday by Senate Majority Leader Mitch McConnell to temporarily shelve a vote on the Republican Obamacare overhaul gives him a few extra weeks to build support for a revised bill before it risks becoming hopelessly stalled by the opposition.

The seasoned GOP leader will be aided by what amounts to a $200 billion piggy bank to push Republican holdouts over the line. That’s the bill’s extra savings compared with the House version that McConnell can tap to provide perks to individual senators, from more opioid assistance to expanded tax-free health savings accounts.

A similar strategy — delay and enticements — worked well in the House, where Republicans last month passed their health care bill on the third try.

But prolonging the debate also gives Democrats and other critics time to mobilize, and ensures that senators will be exposed to an onslaught of opposition as they head home for the weeklong holiday break to defend a bill estimated to leave millions more Americans without insurance.

After the delay was announced, President Donald Trump hosted a White House gathering of all GOP senators. But rather than rally them around the bill with the power of the presidential bully pulpit, he struck a surprisingly detached tone.

“This will be great if we get it done,” Trump told senators in the East Room. “And if we don’t get it done, it’s just going to be something that we’re not going to like. And that’s OK.”

McConnell was more optimistic. “Everyone around the table is interested in getting to yes,” he said after the meeting. “We’ve got a really good chance of getting there. It will just take a little bit longer.”

But fallout seemed to only grow after the vote postponement, as senators realized they would have another opportunity to seek changes and insert their own priorities. As many as 10 Republican senators publicly oppose the bill, though it is likely that several — particularly conservatives seeking deeper cuts — will be swayed.

“The Senate health care bill missed the mark for Kansans and therefore did not have my support,” tweeted Sen. Jerry Moran, R-Kan., typically a reliable Republican vote.

More worrisome for McConnell are GOP centrists who say the bill would force painful cuts in Medicaid and hurt low- and moderate-income consumers insured under the Affordable Care Act.

Sen. Shelley Moore Capito of West Virginia and Sen. Rob Portman of Ohio, whose states are heavily reliant on Medicaid, especially for addiction services, jointly announced their opposition as they seek revisions.

“This bill will not ensure access to affordable health care in West Virginia,” Capito said in a statement. “My concerns will need to be addressed going forward.”

Among the most skeptical is Sen. Susan Collins, R-Maine, a moderate who questioned whether revisions would even make a difference to her.

“It’s difficult to see how any tinkering is going to satisfy my fundamental and deep concerns about the bill,’’ Collins told reporters Tuesday.

With just a 52-seat majority and a wall of Democratic opposition, McConnell can afford to lose only two Republican votes.

His problems are not insurmountable, though. The Senate leader faces the same political dynamic that tripped up House Speaker Paul D. Ryan last month as conservatives pressed to fully gut Obamacare, while centrists, especially in states that have expanded Medicaid, tried to preserve the program’s gains in vastly reducing the number of uninsured Americans.

But the Kentuckian has the benefit of $200 billion in additional savings in the Senate draft, and lawmakers are already lining up for their share.

For example, senators reeling from the opioid crisis are seeking an additional $45 billion for addiction services, while conservatives need about $60 billion for their provision expanding tax-free health savings accounts.

Now the question is not just whether Republicans can amass the support to pass the bill amid their own GOP divisions, but whether they want to take on the political baggage of a health care crisis that has been mainly carried by Democrats until now.

“I hate this,” said one Republican senator.

A nonpartisan Congressional Budget Office report says 15 million more Americans would be without insurance in the first year of the bill’s passage, swelling to 22 million over a decade and wiping out the Affordable Care Act’s coverage gains.

Business groups that typically align with Republicans, including the Chamber of Commerce, urged passage.

But Republican senators have been subjected to withering criticism from patient advocates, physicians groups, nurses and hospitals, which are all warning about the devastating consequences of the health care legislation.

Not a single group representing patients, physicians or nurses supports the GOP measure.

Polling shows scant support for the GOP approach to health care, based on surveys on the House-passed bill, even though support for repealing Obamacare remains high among Republican voters.

“There are so many people, millions of people, who would be harmed by this bill,” said Senate Minority Leader Charles E. Schumer of New York, as Democrats gathered on the Capitol steps holding poster-sized photos of constituents who rely on Obamacare.

“This is not just abstract numbers. These are real people,” Schumer said. “This is who we’re fighting for.”

At its core, the Senate bill, like its companion in the House, does not fully repeal Obamacare. Instead, it ends the Obamacare mandate that all Americans must carry insurance and cuts nearly $550 billion in taxes that were imposed on the health care industry and high-income Americans to pay for expanding Medicaid coverage and providing subsidies for private insurance.

The result, according to an analysis from the Tax Policy Center, is a big tax break for the wealthiest Americans. The top 1 percent of households, those earning more than $875,000 a year, would average a $45,000 tax break, the center said. Those earning beyond $5 million would get a break of about $250,000. Meanwhile, middle-income families earning between $55,000 and 93,000 would see a $280 tax cut.

Trump’s role may expand, especially as conservative senators take their concerns directly to the White House. Sen. Rand Paul, R-Ky., said he had a “very warm” meeting Tuesday to share ideas with Trump. Paul is among four conservative senators pushing for changes, one being a provision to let insurers offer low-cost, bare-bones policies alongside those that meet the Affordable Care Act’s stricter requirements.

The internal strife over the bill is taking a toll on the Republican Party. Trump allies in the America First Policies PAC stunned senators by targeting Sen. Dean Heller of Nevada — perhaps the most endangered GOP senator up for re-election in 2018 — with an ad urging him to vote for the bill. He has voiced doubts.

After GOP senators complained about the ads to Trump on Tuesday, the group announced it was pulling them.

One wealthy GOP donor, Texan Doug Deason, told reporters at the conservative Koch network’s summit last weekend that he had informed House Republicans he would withhold campaign donations until they get behind Trump’s agenda — and pass an Obamacare repeal.

Democrats are eager to keep up the pressure on Heller and other Republicans who have expressed misgivings about the bill, saddling them with Trump’s own assessment of the Republican approach as “mean.”

Sen. Richard J. Durbin, D-Ill., questioned whether Trump would engage beyond the rallying he did to get reluctant House Republicans on board and then reward them for passage with a Rose Garden celebration.

“I’m not sure what his role’s going to be in this,” Durbin said Tuesday. “When it comes down to it, he had a kegger party to celebrate a House bill, which two weeks later he called a ‘mean’ bill.”

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(Staff writer Noam N. Levey contributed to this report.)