The Netherlands' 3-0 win over Brazil in the FIFA World Cup's third-place play-off on Saturday was Van Gaal's last game in charge before he takes over at United and, afterwards, Dutch attacker Robben turned down an offer from the 62-year-old to join him at Old Trafford.
However, Robben's strike partner Robin van Persie, who is already on United's books, hinted Van Gaal would be taking at least one Dutchman from Brazil to the Premier League.
"Admittedly, sometimes I know more than you guys," Van Persie told Dutch TV station NOS when asked if other Dutch national team members would be playing at United next season.
After being dubbed Netherlands' best player in their penalty shoot-out defeat to Argentina in the semi-finals by Van Gaal, central defender Ron Vlaar could be an option to move from Aston Villa to United.
Stefan de Vrij of Feyenoord may also be targeted by Van Gaal as United look to replace long-standing central defensive pairing Nemanja Vidic and Rio Ferdinand.
Robben was arguably Netherlands' leading player of the tournament and was clearly one of Van Gaal's favourites but the 30-year-old looks unlikely to leave Bavaria.
"After the match he did ask me to come to Manchester," Robben said.
"I'm very happy with where I am now. I told [Van Gaal] that."
"Van Gaal is the best coach I've had, we've got a very special relationship but I will not make a transfer.
"I'm at the right place at Bayern so that has nothing to do with Van Gaal or United."
Van Gaal has already signed Luke Shaw and Ander Herrera since taking over as United's manager.
