The 34-year-old playmaker is currently without a club after his contract at Manchester United expired at the end of last month.
According to Spanish news outlet AS, Leeds sporting director Victor Orta has made contact with Mata's father and both parties are keen to reach an agreement.
The report adds that the Whites could face competition from Turkish giants Galatasaray and Fenerbahce for the Spaniard's signature, the latter of whom have recently offloaded former Arsenal star Mesut Ozil.
Mata is believed to be open to the prospect of staying in the Premier League after spending the last 11 years of his career with both Chelsea and Man United.
During his seven-and-a-half year spell at Old Trafford, Mata recorded 51 goals and 47 assists in 285 appearances across all competitions, but he was unable to contribute to any goals in his 12 games for the club last season.
Jesse Marsch has already bolstered his Leeds squad with six new signings this summer, with Brenden Aaronson, Rasmus Kristensen, Marc Roca, Darko Gyabi, Tyler Adams and Luis Sinisterra all arriving at Elland Road on permanent deals.