Spurs had initially been linked with Barcelona left back Jordi Alba, but have now turned their heads towards Italy in their search for a new defender.
Rodriguez has recently been involved in the World Cup with the Switzerland national team, who were only knocked out of the competition on Tuesday, losing 1-0 to Sweden in the last 16.
The news outlet claims that in the 'past few hours', Spurs have made contact with the player, however the Rossoneri are unlikely to want to sell as the player is tied down to the club until 2021.
The report also claims that should AC Milan lose the player, they have identified Fiorentina left back Cristiano Biraghi as a potential replacement for the Swiss defender.
Rodriguez has had a consistent season for AC Milan, featuring in 34 Serie A matches and also appearing in seven Europa League games, managing to grab two goals and two assists in the competition.
The player has also taken part in every minute of Switzerland's World Cup campaign, which comprised of three group stage games and one knockout match.
Tottenham will be hoping the player can replace Ben Davies and Daniel Rose, as Pochettino hopes to finish in top four again next season, and perhaps even challenge for the title.