"We know the sum we are facing as of today," the source said, ahead of a U.S. federal court hearing at which the two sides are expected to report "significant progress" towards reaching a final agreement by late June, according to two sources briefed on the matter.

In April, Volkswagen agreed a framework settlement with authorities in the United States to buy back or potentially fix about half a million cars, after admitting it masked the level of pollution from its cars using cheating software, plunging the carmaker to a 4.1 billion euros operating loss for 2015.

The Wolfsburg-based company could still face further costs, including from a DoJ investigation that could lead to criminal charges, and potential class-action suits.

(Reporting by Jan Schwartz, writing by Edward Taylor; Editing by Georgina Prodhan)