English publication 'The Daily Mail' reports that the 25-year-old is set to team back up with Spurs boss Mauricio Pochettino, who also managed the goalkeeper duing his time at St Mary's.
Gazzaniga will provide Spurs with further back up to first-choice goalkeeper Hugo Lloris.
The Argentine started his career with Unión y Cultura in his homeland before joining the Saints via spells at Valencia and Gillingham.
He was sent out on loan to second division Spanish side Rayo Vallecano last season where he played 34 games in all competitions, keeping 12 clean sheets.
Gazzaniga has made 21 Premier League appeances for Southampton since joining the club from Gillingham in 2012.