Carrick signed a permanent contract to accept the job back in May, after a strong interim spell in charge at Old Trafford, but his side are currently winless in their last eight games across all competitions.
Antonio Conte and Julian Naglesmann have also been linked with the role, but Howe is first choice, based on his sizeable Premier League experience, after becoming a free agent following his departure from Newcastle United over the summer.
A deal for Howe would not be completely straightforward, as United would need to pay the Magpies around £6M in compensation, as he left St. James' Park with years still left on his contract.
The Red Devils would also need to pay for Howe's coaching staff of Jason Tindall, Stephen Purches and others, with reports from Telegraph Sport indicating that Howe has already stated he will not make the move to Manchester without his backroom team in place.