Mkhitaryan has been in fine form for Manchester United this season.
After a tough start, the Armenia international has hit his stride to fire Jose Mourinho's side to within two points of the top four.
Mkhitaryan scored a fifth goal for United during their 3-0 win over struggling Leicester yesterday.
And Carragher has revealed Rodgers wanted to sign the winger for Liverpool during his time at Shaktar Donetsk, only for Mkhitaryan to move to Borussia Dortmund instead.
Manchester United player ratings against Leicester
Sun, February 5, 2017
Express Sport brings you Manchester United's player ratings from their clash with Leicester City
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David De Gea – 7
Henrikh Mkhitaryan starred as Manchester United eased past Leicester
Henrikh Mkhitaryan celebrates scoring for Manchester United against Leicester
"He was a player who nearly came to Liverpool, probably three or four years ago under Brendan Rodgers," Carragher told Sky Sports.
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"I remember speaking to Brendan Rodgers at the time and he wanted to fill that No 10 position behind Luis Suarez.
"He went to Dortmund in the end and that's where he played there.
Henrikh Mkhitaryan was wanted by Liverpool during his time at Shaktar Donetsk
"I think that's always been his favourite position – you're more involved."
Mkhitaryan excelled in the No 10 role as United beat Leicester yesterday.
The 28-year-old is expected to keep his place in Mourinho's starting XI when United go up against Watford on Saturday (3pm).