The 29-year-old winger has been with the La Liga giants since June 2016 and made 30 appearances in all competitions last season, scoring four goals.
However, the Argentina international has slipped down the pecking order under Diego Simeone this term, with just one league start and six outings in total.
The former Benfica man has been told he can leave during the current transfer window and a host of Premier League clubs are said to be interested.
Saints have been joined by Everton, Southampton, Watford and West Ham in showing an interest, according to Spanish publication Marca.
It may well be that Mauricio Pellegrino is eyeing a loan deal in January as the player has been valued at around £17.5m for any would-be suitors to buy outright.