Rangnick was appointed interim manager at Old Trafford in December after Ole Gunnar Solskjaer was sacked.
But he was unable to improve results in the second half of the season as United finished sixth with their lowest points tally of the Premier League era – 58.
Former RB Leipzig boss Rangnick was expected to move into a two-year consultancy role with United this summer.
However, two weeks after the end of the campaign it was announced that he would leave the club permanently to focus on his role as head coach of the Austrian national side.
The announcement came shortly after United confirmed the appointment of former Ajax manager Ten Hag as Solskjaer's permanent replacement.
Sources have told ESPN that Rangnick agreed to cut ties with United in part because of Ten Hag's reluctance to work with the German in his proposed consultancy role.
It is claimed United instigated the termination of their agreement with Rangnick after consultation with Ten Hag.
Rangnick is said to have already been unhappy at not being granted a formal handover with the new manager.
According to sources, Rangnick had expected to meet Ten Hag for a face-to-face meeting but the Dutchman instead opted for a phone call.
The report claims that United bosses became "increasingly exasperated" with Rangnick's comments to the media in the closing weeks of the season, particularly his claim that the club needed to sign 10 new players this summer.
Sources added that United asked Rangnick to sign a non-disclosure agreement as part of his departure from the club.
United confirmed Rangnick’s departure in a statement last month, which read: "We would like to thank Ralf Rangnick for his efforts as interim manager over the past six months.
"By mutual agreement, Ralf will now focus solely on his new role as manager of the Austria national team and will not therefore be taking up a consultancy role at Old Trafford.
"We would like to wish Ralf the best of luck in this next chapter of his career."
United's squad will return from their summer holidays next week as Ten Hag begins his reign as manager.
The Red Devils face rivals Liverpool in their opening pre-season game on July 12 in Bangkok, Thailand.
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