The Spaniard is out of contract at the end of the season, having signed his current deal back in September 2015 when a move to Real Madrid collapsed on transfer deadline day.
Although Man United have the option to trigger a one-year extension in De Gea’s contract, manager Mourinho is beginning to be worried that the 27-year-old could leave for nothing.
A move to Real Madrid now looks very unlikely after Los Blancos signed Thibaut Courtois from Chelsea in the summer.
However, The Times reports that Juventus are hoping to sign De Gea to replace Gianluigi Buffon.
The Italian legend left in the summer after 17 years at the club and they are willing to wait patiently to fully replace Buffon.
Juventus are currently happy with their current goalkeepers, but De Gea on a free transfer would be one of the best deals of all time and the Spaniard could represent the Italian champions for another decade, potentially.
De Gea leaving on a free would be a bitter blow to United and they must tie down their star man to a new contract soon.
Juventus are well known for poaching high-profile players on free transfers. In recent years, that has included Andrea Pirlo, Sami Khedira, Kingsley Coman, Emre Can, Dani Alves and Fernando Llorente.