Germany's Auto Motor und Sport is reporting that the veteran engineer, officially on "holiday" according to his Toro Rosso boss Franz Tost, could be headed to Ferrari.
The report said Ascanelli is currently on holiday on the Adriatic coast.
"Nobody wants to say when he's coming back. Probably never," said journalist Michael Schmidt.
Ascanelli began his career at Ferrari in the mid 80s, and was Gerhard Berger's engineer.
He then moved to Benetton and McLaren, at the latter team working alongside the great Ayrton Senna.
Meanwhile, it is suggested Ascanelli's role at Toro Rosso could be filled by James Key, who departed as Sauber's technical director in February.