SAN FRANCISCO — As Volkswagen faced a court-mandated deadline to reach an agreement with regulators on a plan to fix its polluting diesel cars last month, one lawyer for the German automaker mentioned that he had probably never worked harder in his career.

“I’m optimistic that you’ll be able to break your record in the next month,” Charles R. Breyer, the federal judge presiding over the hearing in San Francisco, replied. “That’s my goal.”

Judge Breyer, who is overseeing the class-action lawsuits brought by American car owners, then gave Volkswagen until April 21 to present a proposal as the first step in the litigation. The company admitted in September to installing software on diesel vehicles to cheat on emissions tests, but hundreds of thousands of its cars remain on American roads, spewing excess pollution.

If Volkswagen does not present a plan that is approved by regulators by Thursday, it could conceivably face a trial before Judge Breyer this summer.