Van Gaal formally took charge at United on Wednesday after his role as Netherlands coach came to an end following the nation's participation at the FIFA World Cup.
Kluivert worked as Van Gaal's assistant at international level for two years and also played under the 62-year-old at both Ajax and Barcelona as well as for their country.
And while Kluivert is disappointed to have been overlooked for the number two role at Old Trafford in favour of Ryan Giggs, he has not ruled out the possibility of forming part of Van Gaal's backroom staff at the Premier League club at some point.
"I talked with Van Gaal and he said he would have loved to have given me the opportunity at Manchester United, but there was somebody already in my position," the legendary former striker told talkSPORT.
"I'm fine with it.
"I would have loved to come to Manchester United. I hope it can happen in the future, you never know."
The synergy that exists between the pair means Kluivert is well placed to make predictions about how United will fare with Van Gaal.
Although returning the club to the UEFA Champions League following the disappointment of last season is arguably the Dutchman's primary aim, Kluivert believes Van Gaal is setting his sights higher.
He added: "He's absolutely going for the Premier League title. A club like Manchester United always has to go for the title and that's what he wants in his first year.
"He's a great coach and I know he's looking forward to taking over at Manchester United.
"It will be a difficult year but I know he can do it. He can work well with the squad.
"Van Gaal is a great tactician and he wants to win matches. He likes to play offensive and attractive football as well though. He won't disappoint the Manchester United fans."
