Although Ms. Trump has not spoken out publicly for action to combat climate change, proponents and opponents of such action see her as an ally. Former Vice President Al Gore met with her during the Trump transition, and was ushered in by the “first daughter” to see the president-elect. The actor and activist Leonardo DiCaprio even slipped her a DVD copy of his climate-change documentary.

“President Trump Must Not Wobble on Climate Change — No Matter What Ivanka Says …,” blared a Breitbart post on Monday written by James Delingpole, who is close to Mr. Bannon and who leads the website’s coverage of climate-change policy.

Mr. Trump wants to make a decision by next week, say people familiar with the White House’s debate on the climate pact, in order to announce his executive order to undo Mr. Obama’s climate regulations in conjunction with his plans for the Paris deal.

According to leaked budget documents, the president will also propose killing off nearly two dozen E.P.A. programs, including the Obama-era Clean Power Program, climate partnership programs with local governments, Energy Star grants to encourage efficiency research in consumer products and climate-change research. Those would be part of a broader budget submission that would cut the E.P.A.’s funding by 25 percent, to around $6.1 billion from $8.2 billion, and its staff by 20 percent.

“If the goal is to fulfill the president’s campaign promises and implement his agenda, there is no value in staying in Paris,” said Thomas J. Pyle, an adviser to the Trump transition and the president of the Institute for Energy Research, an organization partly funded by the billionaire brothers Charles G. and David H. Koch, who have worked for years to undermine climate-change policies.

Mr. Trump has cited Mr. Pyle’s group as being influential in shaping his energy and climate proposals, including his campaign pledge to withdraw from the Paris deal.

“The two greatest obstacles to a Clexit (climate exit from U.N. Paris agreement) are probably Ivanka and Tillerson,” wrote Marc Morano, a former Republican Senate staff member who now runs Climate Depot, a fossil-fuel-industry-funded website that promotes the denial of climate science, in an email. “Tillerson with his ‘seat at the table’ views could be biggest proponent of not withdrawing the U.S. from the agreement.”