The 18-year-old winger has burst onto the scene in Amsterdam to suggest that he is capable of following in the illustrious footsteps of his father Patrick.
A move to Barcelona has been mooted for the youngster – such as the one his dad made in 1998 – with Lionel Messi reported to be a big fan of an exciting talent.
Jose Mourinho is also closely monitoring his development, but Koeman is hoping to see contract talks with Ajax reach a positive conclusion and keep Premier League suitors waiting a little longer.
The former Everton and Southampton manager said ahead of a friendly date against England which could see Kluivert make his senior international debut: "Let's hope these young players stay in Holland a little longer to play every week and develop.
"That's the best [way] but it is not my final decision. I spoke to him and he is in talks with Ajax about his contract.
"I think the best [thing] is for him to stay in Holland because he is only 18 and in my opinion, that is too quick to move to the Premier League.
"It looks like he is not under pressure, he is playing like he did when he was 14 or 15, although he was a little nervous when he was selected for the national team. It's a big talent we have in Holland and let's hope he has a big future."
Kluivert has taken in 24 Eredivisie appearances for Ajax this season, netting seven goals and providing four assists.