The 53-year-old Portuguese, who was sacked by the Blues in December, is increasingly hopeful of succeeding Louis van Gaal at Old Trafford.
No agreement has been reached, and the club have made no comment, but negotiations have begun.
Dutchman Van Gaal, 64, is expected to leave United at the end of the season – one year earlier than scheduled.
That would leave the way clear for Mourinho, who is known to be excited by the prospect of taking over at Old Trafford.
The United hierarchy are thought to be keen to respond after Manchester City secured the services of coveted former Barcelona coach Pep Guardiola earlier this week.
Mourinho’s arrival would mean a resumption of a rivalry that began in Spain when he was coach of Real Madrid between 2010 and 2013.
Twice in recent weeks, Van Gaal’s tenure has seemed perilous.
However, defeats by Stoke on Boxing Day and Southampton on 23 January have been followed by an improvement in both form and performance.
United, who are five points adrift of the Champions League places, play Chelsea on Sunday (16:00 GMT).
Mourinho has never hidden his admiration for United, and particularly their legendary former manager Sir Alex Ferguson.
Speaking before taking his Inter Milan side to Old Trafford for a Champions League last-16 tie in 2009, he said: “I want to face the best and United are a team of champions with a super coach in Alex Ferguson.”
But after being linked with the manager’s job following the departure of Ferguson in 2013, he declared his allegiance to Chelsea.
“I would have turned down every job in the world – the Manchester United job, every one – for Chelsea,” he said.
There were reports last month that Mourinho had written a six-page letter outlining his suitability for the United role, but that was dismissed as “absurd” by his agent. Courtesy:bbc.