The former midfielder arrived at United's training facility on Tuesday and is said to be signing a contract that runs until the end of the season, a source told the AP news agency.
Former England assistant manager Steve Holland will also join the coaching staff, according to the source.
Carrick has previously been an assistant coach at Manchester United and was in charge for a short period when Ole Gunnar Solskjær was fired in 2021. The Norwegian has been interviewed to succeed Amorim, but United now appear to have chosen his assistant as the new coach instead.
Carrick's only long-term tenure as head coach was for Middlesbrough, 2022-2025.
Manchester United are currently seventh in the Premier League and were knocked out of the FA Cup this weekend.
