Audi admitted to having used illegal defeat devices to manipulate its cars' emissions in November 2015; Stadler has been CEO since 2007.
Stadler's arrest is the most high-profile in the diesel emissions scandal.
The Munich prosecutors said Stadler's arrest was not made at the behest of US authorities.
"For Mr Stadler, the presumption of innocence continues to apply", a spokesman said in a statement.
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Victory in Germany cements Federer as the World Number One ahead of the defence of his Wimbledon title on the grass of SW19. I will have a quiet week before Wimbledon with not many things going on, which is great.
Audi CEO Rupert Stadler attends company's annual news conference in Ingolstadt, Germany March 15, 2018. Herbert Diess was given the top job in April and he said that besides focusing on new technologies, like electric cars, he wanted to build a more open, values-based culture to avoid the cheating that led to the emissions scandal.
Volkswagen has pleaded guilty to the criminal charges pressed against the company where two managers are serving prison time in US.
"Audi CEO Rupert Stadler was provisionally arrested this morning", the company said in a statement. The executive was arrested at his home in Ingolstadt in the early hours on Monday, they said. The prosecutors also said that the arrest of Stadler was essential as the investigators saw the risk of him trying to suppress evidence in the diesel emissions probe.
Audi and VW confirmed the arrest and reiterated Stadler was presumed innocent unless proved otherwise.
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Kilimnik worked with Manafort in Ukraine during that time , and during the 2016 election they were reportedly in regular contact. Konstantin Kilimnik was recently and newly named along with Manafort in the attempted witness tampering indictment .
Prosecutors in Munich, Stuttgart and Braunschweig are continuing their investigations of the auto maker and its units.
Volkswagen declined 2.2% to €157.88 and traded 2% lower at 11.50am in local trading, extending losses this year to 5.2%.
In the Reuters report, Stadler had his contract extended in 2017 and was named as a suspect alongside another unnamed member of Audi's board of management.
Volkswagen admitted that almost 600,000 cars sold in the U.S. were fitted with "defeat devices" created to circumvent emissions tests.
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But investigations are continuing elsewhere.