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Rangnick was appointed as the new manager of Austria last week, meaning his consultancy role will be limited at Old Trafford.
And Carragher insists he should NOT be working 'part-time' for Man United while managing Austria.
He described the situation as 'nonsense' as he ripped into the club and the veteran German manager.
Speaking on Monday Night Football, he exclaimed: "That is wrong. That should not be happening.
"The manager of Austria should not be working part-time for Man United, that is a nonsense."
The Liverpool legend continued by claiming it's 'one or the other' regarding the two roles.
"Let's not forget that Rangnick isn't really a manager. He's a director of football or someone who helps the coaches or builds a club," he added.
"And that's the talk about the role he's going in and I just don't get that.
"I don't think you can be the manager of Austria and work for Man United.
"It's one or the other."
Rangnick, who's set to be replaced by Erik ten Hag at the end of the season, will take charge of his first Austria matches in June.
He has UEFA Nations League games against Croatia, Denmark and France on deck following the end of the Premier League season.
He'll also oversee Austria's Euro 2024 qualification campaign and should he secure a spot in the competition, then his contract will be automatically extended by two years.
Sixth-placed Man United, meanwhile, face Brighton and Crystal Palace in their final two league fixtures of the season.