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,” Breyer said, without elaborating on what was left to be done.

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. The company previously reached a nearly $15 billion deal for 475,000 2-liter diesel cars also programmed to cheat on emissions tests.

Hinrich J. Woebcken, president and CEO of Volkswagen Group of America Inc., said the agreement announced Tuesday was part of Volkswagen’s “efforts to make things right” for its customers.

“We are committed to earning back the trust of all our stakeholders and thank our customers and dealers in the United States for their patience as the process moves forward,” he said in a statement.

The deal protects the environment “by removing the cars from road and by offsetting harmful emissions that resulted from their cheating,” said Giles with the EPA.

It requires the company to buy back or terminate leases on 2009-2012 Volkswagen Touareg and Audi Q7 diesels. The EPA says it’s unlikely those can be fixed to meet clean-air standards.

For newer 3-liter vehicles — such as the 2013-2016 Volkswagen Touareg, 2013-2015 Audi Q7, 2013-2016 Porsche Cayenne, and 2014-2016 Audi A6 quattro, A7 quattro, A8, A8L and Q5 — a fix is possible, so buybacks won’t be offered.

Giles said the fix should be submitted soon.

The new settlement appears to mirror the terms for the cars with smaller engines. The previous deal gives 2-liter owners the option to have the automaker buy back their vehicle or pay for repairs.

Volkswagen also will pay those owners $5,100 to $10,000 each.

The company has agreed to spend up to $10 billion compensating those consumers. That settlement also includes $2.7 billion for unspecified environmental mitigation and $2 billion to promote zero-emissions vehicles.

The settlements emerged out of lawsuits from car owners and the U.S. Department of Justice after the EPA said Volkswagen had fitted many of its cars with software to fool emissions tests.

The software recognized when the cars were being tested on a treadmill and turned on pollution controls. The controls were turned off when the cars returned to the road. The EPA alleged the scheme let the cars spew up to 40 times the allowable limit of nitrogen oxide, which can cause respiratory problems in humans.

The company has reached a separate $1.2 billion deal with its U.S. dealers and is still facing potentially billions more in fines and penalties and possible criminal charges.