WASHINGTON — Democrats struggled on Tuesday to salvage a major initiative to address climate change in their sweeping social and environmental policy bill amid opposition from Senator Joe Manchin III of West Virginia, a key Democratic holdout.

Mr. Manchin, a centrist from one of the top coal- and gas-producing states, has pushed to remove or weaken a provision that would impose a fee on emissions of methane, a powerful planet-warming pollutant that leaks from oil and gas wells. He has already effectively succeeded in stripping the bill of its most powerful climate change provision, a program that would have replaced coal- and gas-fired power plants with wind and solar power.

“Senator Manchin has expressed reservations” about the methane fee, Representative Steny H. Hoyer of Maryland, the majority leader, said on Tuesday. “We want to mitigate methane, but we’ll see.”

With House leaders pressing for a vote as soon as Thursday on the sweeping $1.85 trillion social safety net and climate measure, Democrats were rushing to resolve their remaining disputes on the bill and ensure that it would have the votes necessary to pass. Given the evenly divided Senate, the legislation needs the support of every Democrat, making Mr. Manchin’s objections — which he warned on Monday were still outstanding — potentially fatal to the package.

Mr. Manchin has doubled down on his resistance to the methane fee and other major elements of the bill as President Biden attends a climate change summit this week in Glasgow, where he hopes to make the case to world leaders that the United States, historically the largest emitter of planet-warming pollution, is on the brink of enacting a major climate change law. Central to his case is the pending social policy bill, which includes about $555 billion for programs to fight global warming.

Mr. Biden has pledged that the United States will reduce its emissions 50 percent from 2005 levels by 2030. Analysts have concluded that passage of the spending bill would put the United States about a third to half of the way to that goal.

In a speech to world leaders in Glasgow on Monday, Mr. Biden said that 70 countries had joined a coalition led by the United States and the European Union to cut global methane levels at least 30 percent by 2030.

Methane is the second most abundant greenhouse gas after carbon dioxide, and it is responsible for more than a quarter of the warming the planet is currently experiencing. It dissipates from the atmosphere faster than carbon dioxide but is more powerful at heating the atmosphere in the short run.

On Tuesday, the administration announced a major new regulation that would require oil and gas producers to plug their methane leaks. But at the same time, Democratic leaders in Congress want to enact a new law requiring companies to pay the federal government for their leaks of excess methane.

The proposed fee is intended to raise revenue and lower greenhouse gas pollution. Oil and gas producers have complained that imposing a regulation is unfair and duplicative. And they fear it could send energy prices soaring as natural gas prices hit their highest levels in five years.

Climate policy experts say that the double-pronged approach is necessary to shut down methane emissions, particularly because executive regulations could be undone by a future administration.

“It’s important to go after methane from all directions, because it’s been neglected compared to other greenhouse gases, and frankly, we don’t know which of these policy levers will stand up through time,” said Rob Jackson, an earth scientist at Stanford University who has published studies on the climate-warming effects of methane. “Executive actions can be rolled back.”

The proposed fee would apply to the largest oil and gas companies, which emit more than 25,000 tons of greenhouse gases each year. Those companies would pay $900 per ton of leaked methane starting in 2024, ramping up to $1,500 per ton from 2026 through 2030.

Democrats have already lowered those fee levels in an effort to appease Mr. Manchin, according to two people familiar with the private discussions who spoke about them on the condition of anonymity. Staff members crafting the proposal initially set the fee to start at $1,500 per ton of leaked methane, but they lowered it in hopes of winning the senator’s support.

“We have the technology. If the technology is there, we should be using it,” Mr. Manchin said when asked on Tuesday about his position on the methane fee. “We have some issues, and we’re working through all of it.”

According to a report by Energy Innovation, a research organization, the program would eliminate the equivalent of 172 million tons of carbon dioxide from the atmosphere — a similar amount to taking about 36 million cars off the road for one year. The program, which would cost about $75 million to operate, is also estimated to bring in several billion dollars of federal revenue, although the Congressional Budget Office has yet to determine the exact amount.

The measure also includes another $700 million in direct spending to help companies retrofit or employ technology to block emissions and leaks of methane.

Emily Cochrane contributed reporting.