Despite his own reported intentions to stay in LaLiga, De Jong has been linked to a reunion with Erik ten Hag - his former boss at Ajax – with Barcelona attempting to break even on an €85million package.
The 25-year-old has struggled to find a role that best suits him since arriving at Camp Nou in 2019, with Sergio Busquets still starting as Barcelona's deepest midfielder.
Kluivert believes De Jong would flourish in Busquets' role but it would be hard to supplant the veteran, but despite this and Barcelona's recent financial troubles, they should try to keep him.
"I hope that they will keep him," Kluivert said. "If they sell him it is clear that the club would receive a lot of money, but it is a long season and Frenkie is not just any player. I wouldn't sell him.
"Busquets is there, who has had an impressive year and it is difficult to remove him but Frenkie can play in various positions. For me, the best is the 6, but Busi's there."
The 45-year-old scored 91 times in 192 LaLiga appearances for Barcelona, while he remains an ambassador for the club and his son Shane plays in the youth setup.
Kluivert said that with the likes of Ansu Fati, Pedri and Gavi, Barcelona's future can be bright under Xavi, who he believes needs time after replacing Ronald Koeman during last season.
"They are fantastic, I know them very well, they have grown up in the club and they know the values," he said. "Ansu is a pity, for injuries this year but he scores important goals for the club. If he is at 100 per cent we are going to enjoy him at Barça. Pedri has not grown in the club but he is a player of another level.
"Xavi, he is the ideal, but you have to give him a little time. Entering how he has entered in a difficult situation, he did very well.
"I don't know if they were fair to Koeman, but they are situations that happen, and I don't know everything that happened, I can't comment. I didn't speak to Koeman when he came out."