According to John Richardson of the Sunday Mirror, the two clubs have been discussing a potential £10 million deal with add-ons for the summer.
Richardson suggested United manager Jose Mourinho is a big fan of the 18-year-old, who has showcased glimpses of tremendous promise.
"Mourinho has known the teenager since he was a little boy, when he would come to Barcelona with his dad, Dutch star Patrick," Richardson noted. "In those days, Mourinho was a coach at the Nou Camp and has taken a special interest in Kluivert junior's development."
As Richardson reported, after United beat Ajax in the 2016-17 UEFA Europa League final, Mourinho spoke directly to the player.
The winger is said to be excited at the prospect of joining United and playing in the Premier League.
While Kluivert was used sporadically last season, this term we have seen what he can offer with increased regularity. Grup 14's Rafael Hernandez predicted a number of big clubs would be soon making moves to sign the Netherlands under-21 international:
Comparisons with his father, who won the UEFA Champions League with Ajax as a teenager, are natural. But unlike Patrick, who was a centre-forward, Justin is a winger.
Typically stationed on the left flank, the youngster can be such a threat. Kluivert is quick across the ground and relentless in the way he attacks full-backs, and he marries that dynamism with a cool head, meaning he picks passes intelligently and knits attacks together.
There's also a streak of genius in the forward's play. The Ajax Twitter account recently gave us a reminder of what Kluivert can do:
The prospect of Kluivert joining United at such a young age is fascinating, especially considering United are well-stocked at the position in which he tends to operate.
At the start of the season, Mourinho had Anthony Martial, Marcus Rashford, Jesse Lingard and Juan Mata to fill the wide berths. After January, he has Alexis Sanchez to call on too.
With that in mind, it's not clear where Kluivert would fit in, especially if United were to keep all of the aforementioned stars beyond the summer. But as noted below, he is already making his mark at a high level:
For the price mentioned, Kluivert's signing would represent a major coup, as he boasts the talent to go to the top of the game. However, the player would surely have some reservations about making a huge move so soon.
After all, Ajax have a tremendous reputation for nurturing young talent. Another season in Amsterdam—at least—would give Kluivert a brilliant environment in which to keep developing; at United, with competition for places so fierce in attack, it's unlikely he would get the minutes he needs to kick on.