FRANKFURT — Volkswagen is expected to offer cash to the owners of diesel cars in the United States this coming week as it steps up an effort to recover some of the good will 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 on 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 another $500 to $750 that they could spend at a Volkswagen dealer.

The company faces 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.