Based on information received from sources in the company, the newspaper reported that VW may have to buy back around one-fifth of the affected cars in the US, which totalled around 580,000.
The German carmaker, however, refused to comment on the information.
The newspaper said the buy-back could take the form of either a cash payment or the owners would be offered a brand new vehicle in exchange at a heavy discount.
The US authorities would reach a decision on this in January, the report added.
VW is currently engulfed in a scandal of global proportions after it was forced to admit in September that it installed pollution-cheating software in 11 million diesel cars worldwide.
Earlier this week, the US government announced it is taking VW to court over the affair, dubbed 'diesel-gate', which has hit the company's sales and could cost it countless billions of euros in fines and lawsuits.
In Europe, VW is scheduled to start recalling some 8.5 million affected vehicles in the region in January.