According to Reuters , the automaker said, following a far-reaching legal investigation, it had concluded Winterkorn and Stadler had breached their duty of care, adding it had found no violations by other members of the management board.
Winterkorn and Stadler have both denied being responsible for the scandal.
VW in 2015 admitted using illegal software to rig diesel engine tests in the United States, sparking Germany's biggest corporate crisis, the news agency noted. The scandal has VW over EUR32bn (US$38bn) in fines, refits and legal costs.
Winterkorn resigned as CEO on 23 September, 2015, a week after the scandal broke.
About three years later, VW terminated Stadler's contract as Audi CEO against the backdrop of a criminal investigation into whether he was involved in emissions cheating.
According to Reuters, VW said the investigation it launched into the scandal, handled by law firm Gleiss Lutz, included the screening and review of 1.6m files and more than 1,550 interviews and questionings.
"Both Winterkorn and Stadler accomplished great things with the Volkswagen Group … there is no question that the impressive achievements in their professional careers still stand," Volkswagen's supervisory board said in a note sent to staff on Friday cited by the news agency.
"However, as successful as their work was, there were aspects that Winterkorn and Stadler as group board members did not monitor carefully enough," it said in the note, seen by Reuters.
As a result, the company had decided to assert claims for damages against Winterkorn and Stadler on account of breaches of duty of care under stock corporation law, it said.
Winterkorn's lawyers said in a statement cited by Reuters the former CEO regretted the supervisory board's decision and rejected the accusations against him.
"Winterkorn is aware that the supervisory board is obliged to assess potential claims and to possibly assert them. He will therefore seek to clarify those questions in consultation with Volkswagen," the statement said.
Stadler's lawyers declined to comment.
Volkswagen concluded that Winterkorn had breached his duty of care by failing to fully and swiftly clarify circumstances behind the use of unlawful software functions in some diesel engines sold in the North American market between 2009 and 2015.