Hart spent last season on loan at Torino after Manchester City boss Pep Guardiola told him he would not be part of his plans at the Etihad.
The England No.1 had an up and down year in Serie A and has been heavily linked with a move away from the club he's been at since 2006.
And Mourinho is seemingly keen on upsetting his rivals' fans by signing one of their favourite stars.
The Red Devils could lose David de Gea to Real Madrid this summer, with Los Blancos keen to reignite a deal which initially fell through back in the summer of 2015.
Joe Hart odds: Who will sign the Man City keeper? Mon, May 29, 2017
Joe Hart says he won't return to Torino, but which teams are most likely to sign the England star?
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Crystal Palace – 14/1
Jose Mourinho is interested in Man City keeper Joe Hart
But Jose Mourinho is asking the question, is monitoring Joe Hart, in case David de Gea goes
Jonathan Northcroft on Jose Mourinho's interest of Joe Hart
Sunday Times journalist Northcroft says Manchester United manager Mourinho knows it will be difficult to prise a player away from a direct rival but he has enquired into his availability.
"I haven't seen much of him in Italy, I have followed his career there," Northcroft said on BBC Radio 5 Live.
"It hasn't been the season Joe Hart would have wanted but there is still interest there from the Premier League in him.
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Joe Hart is set to leave Man City this summer
"Mourinho likes him as a goalkeeper, he doesn't think he will get him.
"But he is asking the question, is monitoring him, in case De Gea goes.
"The likes of Everton we know have an interest in Joe Hart."