The 53-year-old Portuguese has been the long-time favourite with British bookmakers to succeed Louis van Gaal in the Old Trafford hot seat.
Mourinho, who has also coached Real Madrid, Inter Milan and Porto, has been out of work since being sacked by Chelsea, for the second time in his career, when the 2014-15 Premier League champions were one point above the relegation zone in December.
The Londoners have since climbed to ninth in the table under interim manager Guus Hiddink and play their final game of the season at home to newly-crowned champions Leicester City on Sunday.
Van Gaal said on Friday that he expected to serve the final season of his Manchester United contract he signed in 2014.
“I have said I shall be here, that is my opinion, so the board has to decide if it is like that,” the Dutchman told reporters.
United, who are fifth in the league, end their season with a home game against Bournemouth on Sunday.