With Mr. Piëch’s influence diminished, Mr. Winterkorn pushed ahead with a strategy of improving profit margins at Volkswagen, which continues to struggle to make inroads in the United States and whose growth has stalled in emerging markets like Brazil, China and Russia. The plan includes the discontinuation of unprofitable VW models and the slashing of 5 billion euros, or about $5.5 billion, in operating costs by the end of 2017.

Just two weeks ago, a steering committee of the board had voted unanimously to extend Mr. Winterkorn’s mandate, which was due to finish in 2016, until the end of 2018.

Volkswagen’s traditional culture of highly centralized decision-making could make it difficult for Mr. Winterkorn to deflect suspicions that he and other senior managers at the company’s headquarters in Wolfsburg, Germany, were aware of the software manipulations at the heart of the scandal, experts say.

“This wasn’t a small engineering decision that slipped by management,” said Jo-Ellen Pozner, an assistant professor at the Haas School of Business at the University of California, Berkeley. “It seems to me like something had to be approved by at least a division head.”

With an annual salary of more than €16 million, Mr. Winterkorn is Germany’s highest-paid chief executive. Since taking the helm in 2007 — before the software began appearing in the 2009 model year diesel cars — he has enjoyed the support of Chancellor Angela Merkel as well as the automaker’s powerful workers’ council. But on Tuesday, it was not clear whether Mr. Winterkorn, whom colleagues refer to by the nickname Wiko, would be able to count on their continued support.

“If it emerges that Winterkorn was involved in the issue, then he would step down on his own,” Bernd Osterloh, a Volkswagen board member and labor leader who has until now been an ally of Mr. Winterkorn, told reporters in Frankfurt on Tuesday. “We can’t afford such reputational damage.”

Mr. Winterkorn has nurtured a passion for cars since his childhood in Leonberg, in southwestern Germany, home to a Porsche test site where he used to watch the German sports car maker’s autos whiz by on a track near his home.