"Ferdinand Piech has incriminated the group's former chief executive Martin Winterkorn with a detailed statement to prosecutors," the German magazine said, without citing its sources.
According to the report, Piech told prosecutors that he himself learned from an informant in February 2015 that the company had a "big problem" in the United States.
The source told Piech that US authorities were looking into its use of manipulating software to dupe pollution tests and had passed on their suspicions to Volkswagen.
Piech then asked Winterkorn about it, who assured him that no such document from US officials existed, according to Der Spiegel.
Volkswagen admitted in September 2015 that it had installed so-called defeat devices in 11 million diesel engines worldwide to make the cars seem less polluting than they were.
Winterkorn resigned days after the admission but has always insisted he knew nothing of the scam before it became public knowledge.
Prosecutors in the German city of Brunswick however announced last week that they were investigating Winterkorn for fraud, saying they had "sufficient indications" he knew about the cheating earlier than he has admitted.
According to Der Spiegel, Piech gave his damning testimony when he was questioned by the prosecutors last year.
The spokesman for the prosecution's office in Brunswick could not immediately be reached for comment.
If confirmed, it would be the latest shot across the bow in a long-running rivalry between Piech and his one-time protege.
Piech unexpectedly resigned as head of the German auto giant's supervisory board in April 2015 following a bitter power struggle with Winterkorn after the pair apparently fell out over Volkswagen's difficulties in making inroads in the lucrative US market.
Piech, 79, is the grandson of the inventor of the iconic Beetle, the model on which VW's fortune was built, and was himself VW's chief executive between 1993 and 2002.
He is also one of the representatives of the Porsche family, whose holding company Porsche SE holds 52 percent of VW.