Mourinho has been strongly linked with the Old Trafford post since being sacked by Chelsea in December and the BBC and Manchester Evening News claim discussions have started with a view to the Portuguese coach taking over at the end of this season.
Van Gaal has endured stinging criticism from former United players and the club's fans for his conversative tactics, with reports suggesting he was twice talked out of quitting by executive vice-chairman Ed Woodward earlier this term.
The 53-year-old is believed to prefer to take over in the close-season when he can start with a clean slate rather than inheriting van Gaal's problems with no chance of changing the squad now the transfer window has shut.
The BBC said no agreement has been reached between Mourinho and United, with the club making no comment.
If ex-Real Madrid chief Mourinho did take the United job it would once again put him in direct competition with former Barcelona boss Guardiola after the pair were arch rivals during their time in Spain.
Mourinho was axed by Chelsea just months after leading them to the Premier League title, with the Blues languishing near the relegation zone following the breakdown of his relationship with several members of the squad.
United have fared little better in recent weeks and are currently 10 points behind Premier League leaders Leicester and five points adrift of the top four heading into Sunday's clash with Chelsea at Stamford Bridge.
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