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About 679,000 cars are included in the U.S. recall, covering certain 2011 through 2018 Jettas; 2015 through 2019 GTIs; 2015, 2016, 2018 and 2019 Golfs; 2012 through 2019 Beetles and Beetle Convertibles; and 2017 through 2019 Golf SportWagens.
All the cars have automatic transmissions, manual hand brakes and keyless entry.
VW said silicate can build up on a shift lever switch, allowing the key to be removed if the lever isn't in park. That could let the cars roll off unexpectedly.
Dealers will add a switch and circuit board to fix the problem. The recall starts on Oct. 11.
A VW spokesman said he's not aware of any crashes or injuries due to the problem.
"Diess would go in right away if he could," an anonymous source said, according to Manager Magazin.
The automaker denied the claims Thursday, saying the report was "without merit."
"The latest media speculations (Manager Magazin) regarding VW investments in Tesla are unfounded," a Volkswagen spokesperson told FOX Business.
The magazine claims Diess believes Tesla's electric technologies and software developers would benefit the automaker. However, Volkswagen's families, the Piechs and the Porsches are reluctant at this time.
Tesla did not immediately respond to comment.
FOX Business’ Rachel Tesler and The Associated Press contributed to this report.