Maignan has been heavily linked with a move away from Lille and is almost certain to leave the club in the summer.
Tottenham have identified him as the potential long-term replacement for Hugo Lloris, something done during Mourinho’s time, but AC Milan appear to be the front runners to secure his signature in the summer.
The Rossoneri have an agreement in place with the goalkeeper and are in talks with Lille over a deal ahead of the transfer window.
But AC Milan are not able to formalise a move until Gianluigi Donnarumma’s future is clear, and according to Italian daily Corriere dello Sport, Roma are also on his trail.
Mourinho wants to bring in a new goalkeeper and the club are considering a number of shot-stoppers ahead of the summer.
And Maignan is one of the goalkeepers Roma are considering making an offer for in the coming months.
But they are behind AC Milan in the race who have done considerable groundwork on a potential deal for him.
He is likely to cost €25m for any club who want to finalise a deal with Lille for his signature.