Dutch winger Kluivert, 18, is rated as one of the most promising young players in Europe and is already being linked with a move away from Eredivisie giants Ajax.
United have held interest in Kluivert, the son of Barcelona and Netherlands legend Patrick, for over a year and Mourinho is keen to secure his signature this summer.
The youngster, who made his international debut earlier in the year, is said to be ready to reject a new deal at Ajax that would have made the highest-paid player at the club.
But the Daily Mirror say Kluivert has been told by Van der Sar that he should stay put and continue his development at the club he has been at since the age of 13.
Van der Sar, who played over 250 games for the Red Devils and won four Premier League titles under Sir Alex Ferguson, has been Ajaxs CEO since November 2016.
Kluivert, who is valued at around £40million, dreams of winning the Ballon dOr and talked up a move to Barcelona earlier in the year.
Barcelona, this is my dream club, the teenage sensation told Goal. In the end I would not mind going elsewhere but Barcelona are the dream.
Thats definitely my goal [to win the Ballon dOr] – I can tell you that thats what I want and nothing less.
I will have to work hard to get there but Im only 18 and still have plenty of room for improvement.
I will definitely work hard for it and that is my goal. Theres more to come from me.