The Spain international confirmed earlier this week that he will leave Athletic Bilbao at the end of the season, but Juventus are keen to bring him in when the transfer window reopens in January.
Llorente could leave on a free transfer next year, but the Serie 'A' champions are hopeful that Bilbao will look to cash in on the striker in January, rather than lose him for nothing next July.
Athletic Bilbao coach Marcelo Biesla has insisted the club will not sell Llorente, but the report suggests there's a growing feeling in Spain that the Arsenal target will be on his way in the New Year.
The 27-year-old played a key role as Bilbao reached the Europa League final last season, only to fall at the final hurdle against Porto.
Reports in Italy, from Italian newspaper Gazzetta dello Sport, claim a move to Juve looks increasingly likely for Llorente, which would come as a blow to Arsenal.
The Gunners appear set to spend in Janaury, with Arsenal manager Arsene Wenger reportedly keen to bring in one or two new faces during the transfer window.
The north London side welcome West Brom to the Emirates Stadium this weekend, looking to improve on their previous showing in front of the home fans - a 2-0 defeat to Swansea that saw them booed off the pitch.