VW looks set to pay $4.3 billion (€4.1 billion) to settle a US criminal case after admitting to fitting out 11 million diesel cars worldwide with software that reduces emissions under testing to pass controls, but then
switches off under real driving conditions.
That meant they released up to 40 times the permitted pollution levels.
Previously, VW claimed that former group chairman Martin Winterkorn was only made aware of the issue in late August to early September 2015, just before the scandal broke out in September 2015.
But newspaper Süddeutsche Zeitung and regional television channels NDR and WDR claimed that two "crucial witnesses" have told US investigators that both Winterkorn and current group chairman Herbert Diess knew about the circumvention software "from the end of July 2015".
"The directors took no measures to inform American authorities about these manipulations", wrote Süddeutsche Zeitung on its internet site.
Of the up to 11 million vehicles affected, 600,000 were in the US alone.
Although Volkswagen admitted to installing the software, Winterkorn denied responsibility.
According to German media, one of the two witnesses was the head of the manufacturer's diesel service and told US authorities about the cheating software in August 2015.
Despite the scandal, VW Group - which includes the brands Audi, Porsche and Skoda - said on Tuesday it had sold a new record 10.3 million cars worldwide last year.
The latest revelations came just 24 hours after a former Volkswagen executive was charged with fraud and conspiracy.
Oliver Schmidt, who ran VW's US regulatory compliance office from 2012 to March 2015, is accused of lying to US regulators.
Volkswagen has already settled civilian charges related to the scandal, agreeing to a $14.7 billion payment that allows nearly 500,000 vehicle owners to sell back their cars or get them fixed.