Volkswagen and US officials have reached a deal under which the car maker could offer to buy back up to 500,000 diesel cars in the US, according to reports.

The German car giant has also agreed a compensation fund for owners.

VW is expected to reveal the deal to a Federal judge in San Francisco on Thursday.

A VW spokeswoman, the Environmental Protection Agency and the Justice Department declined to comment.

Volkswagen could also offer to repair diesel vehicles if US regulators approve a fix at a future date, reports said.

US District Judge Charles Breyer in March gave VW until Thursday “to announce a concrete proposal for getting the polluting vehicles off the road.”

Judge Breyer said in March the “proposal may include a vehicle buy back plan or a fix approved by the relevant regulators that allows the cars to remain on the road with certain modifications.”

In September last year the Environmental Protection Agency found that VW cars being sold in the US had a “defeat device” – or software – in diesel engines that could detect when they were being tested, and change the performance to improve results.

Some models could be pumping out up to 40 times the legal limit of the pollutant nitrogen oxide.

In March, Volkswagen’s boss Matthias Mueller said that a deal with US authorities over its emissions scandal could take longer and cost more than expected.

He warned that the €6.7bn (£5.2bn) set aside to cover the costs of the scandal might not be enough.