Van Gaal has been under intense pressure at Old Trafford following a disappointing campaign. United are facing a fight to qualify for next season's Champions League, having exited the group stage of the competition in December.
They were also knocked out of the Capital One Cup by Championship side Middlesbrough and Van Gaal has faced heavy criticism for his style of play.
Mourinho is available following his sacking by Chelsea in December and it is little secret that he covets the United job.
Scarves bearing his name were even on sale outside Old Trafford at a recent Premier League game as pressure mounted on Van Gaal.
He has come close to being appointed before, and was considered to succeed Sir Alex Ferguson when he left in 2013. On that occasion, doubts over Mourinho's personality and style of football - most notably from United non-executive director and former player Sir Bobby Charlton - led to David Moyes being appointed.
Van Gaal is understood to have offered to resign following a defeat at Stoke on Boxing Day, and was jeered by fans at Old Trafford following the 1-0 loss to Southampton last month.
However, since then he has been more bullish over his prospects of staying at United, who have progressed to the fifth round of the FA Cup and beaten Stoke City in the Premier League in the last week.
“I have signed a three-year contract and I have said in all my press conferences that it’s a process, not one game,” Van Gaal said in December. “And I want to continue until the end. If I am not willing to do that, I don’t think I would have started here 1½ years ago.
“It is fantastic of course, but that also gives you a lot of pressure. When the board has such confidence, the pressure is much higher than when they say it is your last game or something like that.
“When your confidence is not so high, you can fight against it. When you can fight, you have bigger motivation than when they support you. But I am always fighting.”
Van Gaal leaves United
- Telegraph