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De Jong worked under Ten Hag at Ajax and developed into one of the best midfielders in the world in the 2018/19 season, where they had a memorable run to the Champions League semi-finals.
That prompted a €75 million switch to Barcelona, where the Netherlands international has played 136 times.
On his Twitch channel, journalist Gerard Romero says Ten Hag wants to reunite with De Jong but Barcelona believe United may try and arrange a swap deal.
Erik ten Hag has asked for Frenkie de Jong as a priority. Alex Telles, Harry Maguire and Marcus Rashford are players Barcelona think #mufc could offer. Barcelona may be interested in Scott McTominay and Anthony Elanga #mulive [@gerardromero]— utdreport (@utdreport) April 30, 2022
And Maguire is one of three names mentioned as a potential makeweight. The England defender has been widely criticised following a series of poor displays this season and his future as skipper is not clear.
United paid £80 million, a world record fee for a defender, to bring him in from Leicester City in 2019.
Another player Barca think United could propose is Marcus Rashford, who has also had a disappointing campaign.
The academy graduate has looked devoid of any confidence and has scored just five goals in all competitions - his last coming back in January.
Left-back Alex Telles has also been linked, while it's said Barca might be interested in Scott McTominay and Anthony Elanga.
Interim manager Ralf Rangnick, who will become Austria manager alongside his new role as a consultant, has already said United may need as many as TEN new players.
It's claimed Ten Hag is ready to build his first-team around eight players - including youngster Hannibal Mejbri, Alejandro Garnacho and James Garner.
He apparently sees the likes of Bruno Fernandes and Jadon Sancho as central to his plans.