The 25-year-old fell down the pecking order at White Hart Lane last season, but the arrival of Mauricio Pochettino could see "The Beast" instilled back into the side on a regular basis.
Napoli are also believed to be interested in the tough-tackling midfield enforcer, but it is unclear whether Spurs are willing to let him leave.
Garcia's move to Russia should stop AVB from targeting any of the Spurs midfielders he worked with during his tenure at the club, before being sacked last December.
The versatile Spaniard failed to impress during his time in Manchester following his move to the club from Benfica in 2012, and has been moved on as the club had too many foreign players in the squad.
AVB has enjoyed a fine start to life in Russia with Zenit as he has lead the side to 12 wins from 14 matches since taking charge in March.