Volkswagen has reached a deal with the US authorities under which the automaker could offer to buy back up to 500,000 diesel cars in the US.

VW has also agreed a compensation fund for owners.

The German car giant is expected to reveal the deal to a Federal judge in San Francisco on April 21.

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

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

In March, US District Judge Charles Breyer gave VW until April 21 “to announce a concrete proposal for getting the polluting vehicles off the road.”

Judge Charles 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 2015, the EPA 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, VW CEO Matthias Muller said that a deal with US authorities over its emissions scandal could take longer and cost more than expected.

Matthias Muller warned that the €6.7 billion set aside to cover the costs of the scandal might not be enough.