Senator Joe Manchin III of West Virginia has pushed Democrats to drop or weaken a second major climate change provision from the sweeping social policy and environmental spending bill that the White House hopes to finalize this week, according to two people familiar with the matter.

Mr. Manchin, a centrist Democrat from one of the country’s top coal- and gas-producing states, wants to remove or modify 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 rapidly shut down coal and gas-fired power plants and replaced them with wind and solar power.

Democrats are racing to finalize the details of the massive budget bill this week. President Biden is set to attend a major climate summit in Glasgow, Scotland, this weekend where he hopes to point to that bill to make the case that the United States, the world’s largest historical greenhouse polluter, is finally taking strong, forceful action to cut its fossil fuel emissions — and to push other countries to do the same. Mr. Biden has pledged that the United States will reduce its emissions 50 percent from 2005 levels by 2030.

Analysts have found that it will be technically possible, although difficult, for the United States to meet that goal without passing the clean electricity legislation that Mr. Manchin opposes. The broader spending package still includes about $300 billion in tax credits for wind and solar energy, which analysts say could get the United States about halfway to Mr. Biden’s target. But removing the methane fee legislation could further weaken Mr. Biden’s case in Glasgow.

A spokeswoman for Mr. Manchin did not respond to a request for comment.

A spokeswoman for the senator who is writing the methane fee legislation said it has not yet been excised from the bill.

“The methane fee is not out of the package,” said Rachel Levitan, a spokeswoman for Senator Thomas Carper, the Delaware Democrat who leads the Senate Environment Committee. “Chairman Carper is working to get robust climate provisions in the reconciliation bill and is in active negotiations to ensure that the bill meaningfully reduces greenhouse gas emissions.”

Another person familiar with the matter said that Mr. Manchin appeared open to negotiating the details of the methane fee to make it easier and cheaper for natural gas companies to comply.

Separately, the Environmental Protection Agency is expected to release a draft regulation as soon as this week that would compel oil and gas producers to monitor and plug methane leaks from existing oil and gas wells. Among Mr. Manchin’s objections to the methane pollution fee is that it could be duplicative of those rules, according to the two people familiar with the matter.

While Senate Democratic leaders have pledged that the broader budget legislation — which could run between 5,000 and 10,000 pages — will be completed this week, people familiar with the process said it was more likely that Democrats would agree to a broad-brush framework of a deal before Mr. Biden travels to Glasgow, and that he would have to make the case to the world that lawmakers would indeed soon pass the bill.

“The entire world knows the name Manchin now,” said Rich Gold, a Democratic energy lobbyist and former E.P.A. adviser. “So if he is able to go to Glasgow and say, here is the piece of paper representing the deal, and here’s Senator Manchin’s name on the bottom, I think the Germans will be OK.”