Diess issued a statement calling his use of the phrase "definitely an unfortunate choice of words", according to the BBC. "I would like to apologise in any form".
Diess said "EBIT macht frei", which is similar to "arbeit macht frei", a slogan that literally means "work sets you free", and that was engraved on the gates of Nazi concentration camps.
EBIT is an acronym for Earnings Before Interest and Tax, a key indicator of a company's profit.
Herbert Diess used the pun "EBIT macht frei" during a speech at the firm's management gathering on Tuesday.
"It was in no way my intention to put my statement in the wrong context - I honestly didn't think it would at the time", he said.
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Volkswagen was founded in 1937, as part of Nazi leader Adolf Hitler's vision to enable German families to own their first vehicle.
Volkswagen, whose factory was repurposed during World War II to build military equipment and vehicles, is today the world's biggest automotive group with brands including Audi, Bugatti, and Porsche.
Diess added that he, the company and its staff were "aware of the particular historical responsibility of Volkswagen in connection with the Third Reich".
Asked whether Bernstein analyst Max Warburton was right to suggest that Diess had lost support internally as a result of the remarks, Volkswagen's supervisory board said such an inference was inappropriate.