Gazzaniga spent five years on Southampton's books, spending last season in Spain's second tier with Rayo Vallecano where he made 22 appearances. He has now completed a move to north London where he is expected to serve as Tottenham's third-choice goalkeeper behind Hugo Lloris and back-up Michel Vorm.
Spurs had been expected to hand that role to Spanish shot stopper Pau Lopez, who spent last season on loan at White Hart Lane despite not making a single appearance.
Tottenham had an automatic option to sign the 22-year-old but earlier this summer IBTimes UK reported the club were trying to re-negotiate a cut-price deal for the goalkeeper after he entered the final year of his contract with parent club Espanyol.
Espanyol turned down that offer however, demanding €4m for their player. Tottenham decided to withdraw from negotiations with their option to sign him expiring on 30 June.
Tottenham have instead turned to Gazzaniga, who worked with his countryman Mauricio Pochettino during the manager's 18-month spell on the south coast. The Argentine becomes Tottenham's second signing of the summer after the club announced a deal for defender Davinson Sanchez last week.
Spurs confirmed last Friday confirmed the capture of the 21-year-old from Ajax having a struck an agreement over a fee with the Dutch club after some tough negotiations and agreed personal terms with the Colombia international. The move is still subject to a medical and a work permit.
Tottenham are also reportedly close to completing a deal for Celta Vigo midfielder Pape Cheikh Diop. The Premier League side have seen an improved offer said to be worth €10m accepted by the La Liga side and are ready to offer Diop six times what he currently earns in Spain.
Spurs also remain heavily linked with a move for Ross Barkley, who has been told he can leave Everton this summer. Pochettino told reporters last week he is eager to make four additions before the transfer window closes on 31 August.