The Scottish international will be allowed to leave in the January transfer window as Old Trafford bosses wish for him to receive more first-team opportunities.
The length of the deal is not yet known for the player who has made 33 appearances for the Red Devils since breaking through from the youth ranks.
A late substitute in the recent 1-0 win over Tottenham Hotspur, it has been the only time the 22-year-old has featured under interim boss Ole Gunnar Solskjaer.
Celtic manager Brendan Rodgers refused to comment on the speculation when asked about it in Thursday’s press conference, though he did admit that McTominary was a “fantastic talent”.
Rodgers said there would still be ins and outs at Celtic before the end of January. Things have gone a little quiet over the past week following the flurry of activity earlier in the window with attackers Oliver Burke, Timothy Weah and Vakoun Issouf Bayo all arriving.