The north London club are the only team in the Premier League yet to make a single signing in the summer transfer window with the 2017/18 season set to commence in just over two weeks
Sources have told ESPN FC that the Argentine could be the club's first arrival as manager Mauricio Pochettino seeks a replacement for Paul Lopez.
Lopez signed for Spurs last summer in a season-long loan deal from La Liga side Espanyol and served as the club's third choice goalkeeper behind Hugo Lloris and Michel Vorm.
Pochettino now needs a replacement and believes that Gazzaniga, who he worked with during his time as Southampton manager, could be the ideal replacement to challenge Vorm as the second choice goalkeeper.
The 25-year-old has made a total of 21 appearances in the Premier League for the Saints across four seasons, having only served as a second choice goalkeeper and spent last season on loan at Rayo Vallencano in the Spanish Segunda Division, where he made 32 appearances.
The news is not something that will cheer up Spurs fans who have been frustrated at the club's lack of transfer activity. Along with being the only club in the Premier League, they are also the only club in England's top four divisions not to have made a single signing.
However, Pochettino recently assured fans not to be worried and that Tottenham would be making moves in the transfer market soon.
"I want to tell our fans, 'Don't be worried'," Pochettino said. "Because today there are a lot of rumours and stuff about the clubs spending big money and it looks like Tottenham are not ambitious."
"That is not real. We will move in the market, for sure, to try to improve the squad and we will be competitive like the last two seasons. We have a plan. It's early and we're calm because our squad is competitive. If we compare the last two seasons with other teams, we were the best.
"That is why we are calm. We have a very clear idea of what we want and what we need to do to try to achieve our target."