The Netherlands international has been linked with a potential move to Manchester United over the past 12 months or so.
Kluivert featured in Ajax’s 2-0 loss to Manchester United in the Europa League final at Stockholm last season.
The teenager has scored seven goals and has made four assists in 24 appearances in the Eredivisie this term.
Kluivert, who is the son of former Ajax and FC Barcelona legend Patrick, is rated as one of the Netherlands’ next big prospects.
Reports in the British media suggested that Kluivert could move to Manchester United in the summer after the Ajax forward talked up a potential move to Old Trafford.
Marcus Rashford and Anthon Martial have seen their progress stunted at Manchester United since Mourinho’s appointment, although Jesse Lingard and Scott McTominay have flourished under the Portuguese head coach.
Netherlands head coach Koeman has revealed that he has urged Kluivert to remain at Ajax for another season.
“I spoke to him about this,” Koeman is quoted as saying by Metro.
“He’s in talks with Ajax about his contract.
“I think the best is for him stay in Holland because he’s only 18.”
Manchester United signed Chile international Alexis Sanchez in a swap deal from Arsenal in the January transfer window.
The Red Devils will take on Tottenham Hotspur in the FA Cup semi-final.
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