Gennaro Gattuso is currently in charge of the San Siro outfit, but Corriere della Sera believes he will only remain in the post if he leads his side back to the Champions League.
The Rossoneri currently occupy the fourth and final qualification spot, but have dipped in form recently and are only one point ahead of Atalanta.
CEO Ivan Gazidis, who joined Milan from Spurs' rivals Arsenal last year, is understood to have identified Pochettino as the ideal candidate for the job.
Gattuso has another two years left to run on his contract and his sacking would cost Milan around £8.6 million, but the coach has clashed with the club's sporting director Leonardo recently.
The outlet says that Gazidis is attracted by the idea of bringing in a big name from the Premier League, who would be ready to bring the best out of a group of young players.
Pochettino, who has a contract until 2023 at White Hart Lane, has led his side to third place in the Premier League and the quarter-finals so far this season.
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