Citing reports in Italy, the article claims the Spain international is already growing impatient with his place on the Juventus bench. Arsenal and Tottenham have both been linked with the forward, while Chelsea are also thought to be monitoring developments.
The report claims Llorente has asked the Serie A champions to let him look for another club in the winter window if they plan on using him as an impact substitute.
The 28-year-old striker has made just one appearance off the bench for the Italian giants in their opening three fixtures.
And the unsettled Spaniard is being eyed up by a number of Premier League clubs. Arsenal have been long-term admirers - Arsene Wenger reportedly scouted the forward at Athletic Bilbao - while Tottenham are also tracking developments.
Emmanuel Adebayor has been forced to train with the development squad at White Hart Lane, so Jermain Defoe and Roberto Soldado are the club's only first-team forwards.
As well as Arsenal and Spurs, fellow London club Chelsea are also believed to be monitoring Llorente.
Mourinho typically favours a physically imposing forward up front, and was linked with Christian Benteke over the summer.