California has a waiver under the 1970 Clean Air Act to set its own air pollution regulations, and a dozen other states follow its lead. If the Trump administration loosens federal pollution rules, California has vowed to stick with its own stricter standards and to sue the administration.

That fight could, in effect, split the American car market in two and would set up a huge legal battle that is likely to reach the Supreme Court.

The proposal, which is to be jointly released by the E.P.A. and the Transportation Department, was largely completed in May. It was sent by both agencies to the White House for review, after which it was expected to be published in June or early July in the Federal Register, a major step in formalizing a new regulation.

The details of the proposal are largely similar to those in a draft version described by The New York Times in April, according to people who have seen the plan. After its publication, the administration will take public comment before potentially revising and releasing its final plan, expected this year.

While the 11 people familiar with the talks described a bitter feud between Mr. Wheeler and Mr. Rosen over how to proceed, Mr. Wheeler described the negotiations between the agencies as “business as usual.”

“This assertion of a ‘nuclear war’ is categorically false,” he said in a statement. “Our efforts with DOT have been reflective of a robust and constructive interagency process that the American people expect — and deserve — when agencies are proposing rules of such consequence.”