Former England midfielder Carrick played 464 games in all competitions for United between 2006 and 2018, having been signed from Tottenham by Sir Alex Ferguson.
He has been out of work since he was sacked by Championship club Middlesbrough last June after two and a half years in charge.
Carrick guided Boro to the Championship play-offs in the 2022-23 campaign, only for them to lose to Coventry in the semi-finals.
His side then narrowly missed out on the top six in successive campaigns, resulting in Carrick losing his job.
Wallsend-born Carrick does not boast the same managerial experience as Solskjaer, given he has had only one permanent role with Middlesbrough.
Following his retirement from a glittering playing career at Old Trafford, Carrick immediately joined Jose Mourinho’s coaching staff at United at the end of the 2017-18 season.
After the Portuguese was sacked, he was kept on as a first-team coach throughout Solskjaer’s 168-game reign.