United have reached out to Mourinho's representatives since he was sacked by Chelsea in December.
No formal job offer or promises have been made by the club, but further discussions are likely in the close season, One World Sports report.
Van Gaal could yet keep his job.
United need to wins from their final two games to guarantee a top four finish.
And combined with an FA Cup win over Crystal Palace on May 21 could see the veteran coach see out the final season of his three-year contract.
Mourinho turned down a mid-season offer from Real Madrid to take the job at the Bernabeu.
The Portuguese spent three seasons at the club and president Florentino Perez wanted him back.
But Mourinho wants to work in England so told Real he wasn't interested.
The ex-Chelsea and Inter Milan boss is said to be aware of opposition to his appointment amongst top brass at United, with the nine-man board understood to be split on whether his arrival would be for the best.
