The 23-year-old is expected to leave the Championship club this summer after attracting growing interest from several Premier League sides.
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According to The Northern Echo, Carrick has strongly advocated for Hackney during recruitment discussions at Old Trafford, believing the midfielder is ready for the step up to top-flight football.
The pair worked closely together at the Riverside, where Hackney developed into a key creative force over the last three seasons.
Middlesbrough are now preparing for offers after missing out on promotion following a narrow play-off final defeat to Hull City.
With just one year left on his contract, the club are open to selling one of their most valuable assets.