Chelsea have learned that doing a deal for Maignan is a possibility, because the goalkeeper would be interested in a move to Stamford Bridge.
The 29-year-old has been a regular for Milan since joining them on a five-year contract from Lille in 2021.
He has been first choice for France since Hugo Lloris retired from international football in January 2023.
Chelsea have struggled to find a long-term option in goal since buying Kepa Arrizabalaga in 2018 for £71m, which remains the world-record fee for a goalkeeper.
Two years later they brought in Edouard Mendy, who left for Al-Ahli in 2023, while Kepa has spent the past two seasons on loan at Real Madrid and then Bournemouth.
In the past two years the Blues have also signed Robert Sanchez, Djordje Petrovic and Filip Jorgensen but the club appears unconvinced by all three.
Petrovic spent last term on loan at Strasbourg while Sanchez and Jorgensen both had spells in the Chelsea team.