SAN FRANCISCO — Volkswagen reached a deal that will give at least some owners of the remaining 80,000 diesel vehicles caught in the company’s emissions cheating scandal the option of a buyback and provide compensation to all of them on top of any repurchase or repairs, U.S. regulators and a federal judge said Tuesday.

The $1 billion settlement with the U.S. Environmental Protection Agency will give owners of 20,000 3-liter diesel cars the choice of a buyback. The figure does not include additional payments to owners.

Volkswagen believes it can bring the other 60,000 vehicles into compliance with pollution regulations and will not offer a buyback if that’s the case, U.S. District Judge Charles Breyer said in San Francisco.

The deal includes $225 million the German automaker will contribute to an environmental fund to offset the cars’ excess pollution, Cynthia Giles of the EPA said in a conference call with reporters.

Additional compensation for car owners will be substantial, according to the judge, but he did not provide a figure and said the sides still had more work to do.

“I am optimistic the parties will resolve the remaining issues,” he said.

The settlement was a major step toward rectifying lawsuits stemming from the global scandal that erupted last year, damaging Volkswagen’s reputation and hurting its sales.