Volkswagen officials would not confirm a report that said diesel owners would be offered up to $1,250 this week.

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FRANKFURT, Germany — Volkswagen is expected to offer cash to owners of diesel cars in the United States this week as it steps up an effort to recover some of the goodwill it lost after admitting in September that the vehicles were programmed to cheat on emissions tests.

Volkswagen officials said the company would make an announcement Monday but would not confirm a report on an automotive website that diesel owners would be offered up to $1,250.

The site, The Truth About Cars, said the owners would get a cash card worth $500 that they could spend any way they liked, and an additional $500 to $750 that they could spend at a Volkswagen dealer.

The company faces many lawsuits from owners seeking compensation for the decreased resale value of the roughly 500,000 Volkswagen and Audi vehicles that were equipped with illegal software. It was not clear whether owners would have to give up any rights to sue if they accepted the cash.

The illegal software allowed the cars to detect when emissions testing was under way and turn on the emissions controls. But when the cars were on the road, they emitted up to 40 times the permissible amount of nitrogen oxide, a pollutant harmful to the lungs.

Volkswagen has promised to make changes to the affected cars so they comply with emissions limits. But the repairs are likely to be costly and complicated, and Volkswagen has not said how they would be carried out.

Two dealers said Saturday they were aware Volkswagen was planning something but did not know specifics. “There is a program in the works with VW, that I do understand,” said Alan Brown, the head of an association of Volkswagen dealers.

Cars with illegal software include diesel Golf, Jetta, Beetle and Passat models sold since the 2009 model year, and Audi A3 cars.

The Environmental Protection Agency (EPA) said last week that some larger Volkswagen, Audi and Porsche diesel vehicles, all of which are produced by Volkswagen, also had software that was not allowed.

But Volkswagen has denied the software in those cars was designed to cheat on emissions tests, and they would probably not be eligible for the cash compensation.

Already, Volkswagen has been trying to minimize the damage to sales by offering cash incentives to existing owners if they buy or lease a new car, and big discounts for all buyers.

Honda recall

Honda is recalling more than 25,000 cars in the U.S. to replace the rear grab rail brackets, which it says could interfere with the deployment of a side-curtain air bag.

The carmaker says it discovered the faulty design during internal testing.

The company says owners of affected vehicles should go to authorized dealers once they receive notification of the recall. It says notifications will be mailed to customers starting in early December. The recall affects 25,367 of the Fit LX vehicles, model year 2015-2016.