The 20-year-old was a prospect of Ajax's youth setup for 11 years between 2010 and 2021 before the Eredivisie champions allowed him to join Leipzig on a free transfer last year.
Brobbey returned to Ajax on loan for the second half of the 2021-22 season, during which he scored seven goals in 11 Eredivisie matches to help his side win the title.
Leipzig have now welcomed Brobbey back to the club following his time with Ten Hag at Ajax, and he is under contract with Domenico Tedesco's side until 2025.
However, Brobbey has only provided three assists in 13 matches for Leipzig since his arrival, and he is still waiting for his first goal for Die Roten Bullen.
Following his strong temporary stint under Ten Hag earlier in 2022, Brobbey has been tipped to reunite with the Dutchman at Old Trafford amid uncertainty surrounding a number of Man United attackers.
However, Dutch outlet De Telegraaf has claimed that the youngster has personally called Ten Hag to inform him that he has no intention of signing for Man United this summer.
Instead, the report adds that Brobbey will instead return to Ajax on a permanent deal, as the Dutch giants possessed a buyout clause in his loan deal from Leipzig.
Ajax will supposedly fork out €17m (£14.5m) to bring the Netherlands Under-21 international back to the club, and he will sign a five-year deal with the Dutch champions.
Brobbey would arrive back in Amsterdam as a replacement for Sebastien Haller, who will fill the Erling Braut Haaland void at Borussia Dortmund.
Man United have been rocked by the news that Cristiano Ronaldo has supposedly asked to leave the club this summer, with the five-time Ballon d'Or winner not convinced that his side can challenge for trophies in 2022-23.
However, Ronaldo could struggle to secure an exit from the Red Devils due to a lack of suitors, with none of Barcelona, Bayern Munich or Chelsea thought to be considering moves.
Marcus Rashford is expected to stay and try to reignite his Old Trafford career under Ten Hag, but Anthony Martial's future is still shrouded in uncertainty.
Man United have already confirmed two summer signings in the shape of Tyrell Malacia and Christian Eriksen, the latter of whom was announced as a Red Devil on Friday.
Despite seemingly missing out on Brobbey, Man United will soon make one addition from Ajax, as Lisandro Martinez has confirmed his imminent arrival in England.