The Argentina international has been made available by Atletico this month and is a target for Saints boss Mauricio Pellegrino.
However, Sky Italia says Inter are also keen and willing to make a swap offer involving Joao Mario. Inter coach Luciano Spalletti has given his approval to the move.
Pellegrino had hoped to take Gaitan on-loan to the end of this season.
For his part, Atletico coach Diego Simeone is happy to see Gaitan move on as he doesn't have him in his first team plans.