Rangnick led the Red Devils on an interim basis in the second half of the 2021-22 campaign and had agreed to take on a consultancy role at the club, which was expected to last for two years.
However, it was revealed at the end of April that Rangnick would become Austria's new head coach, which put his position at Man United in doubt.
The 63-year-old was initially set to balance the two roles, but it was later revealed that the German would instead leave Old Trafford on a permanent basis this summer.
"We would like to thank Ralf Rangnick for his efforts as interim manager over the past six months," read a statement on Man United's official website.
"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."
According to ESPN, Man United's new head coach Ten Hag was reluctant to work with Rangnick in his proposed consultancy role, while the Dutchman refused to meet the interim manager face to face, instead deciding to speak with him over the telephone.
Rangnick found it difficult to make his mark at the helm, leaving his role as interim head coach with a record of 11 wins, 10 draws and eight defeats from 29 matches, with Man United missing out on a top-four spot in the Premier League table, finishing sixth, which secured a spot in the Europa League for next term.
The German had said that he had discussed his role at the club "in detail" with Ten Hag and was confident that he would be an asset moving forward.
Rangnick had previously acted as head of sports and development for Red Bull, working with both Red Bull Salzburg and RB Leipzig, and he was set to have a role in Man United's summer transfer business.
"We spoke about that in detail," Rangnick said when asked about his role at Man United under Ten Hag's management. "About all the different areas where I could maybe be of help. It's not that easy or not that difficult to know in which areas this could be.
"I think we showed that in the last 15 years with Hoffenheim, Salzburg and Leipzig, even for clubs not as prominent as Manchester United, it's possible to identify, develop and even at one stage sell.
"I know Manchester United is not a selling club but rather a developing and buying club, but this is possible and this is what is most important, that the club finds players for whom it is the next logical step is in their career, to develop their sporting career. If that happens, I can be positive and I can hopefully be encouraging for our fans that we will bring United back to the top."
Man United are yet to make a signing during this summer's transfer window but are said to making strong progress in their bid to bring Barcelona midfielder Frenkie de Jong to the club.
Ajax's Antony is also believed to be a leading target, with the 20-time English champions working to boost their squad ahead of their first game of pre-season against Liverpool on July 12.