Sheedy has been on the Toffees' coaching staff since 2006, and is currently in charge of the under-18s, so has first-hand experience of working under Moyes before he moved to Old Trafford last summer.
However, he was far from complimentary.
"All of you out there Moyes was never interested in our youth team or players," Sheedy, who played 314 times for Everton from 1982 to 1992, tweeted after the Red Devils' 3-0 defeat to Liverpool on Sunday.
"In my 7 years moyes showed no intereset [sic] in our youth team."
Asked on the social media site whether that must have "done his head in", Sheedy replied: "Yep, he did.".
Press Association Sport understands when Everton became aware of those tweets they asked Sheedy to remove them and they were subsequently deleted.
However, his criticism of Moyes' tactics - "Punt the ball up to Fellaini. Great viewing," - in Sunday's defeat remained.
In response to another follower Sheedy, a former Republic of Ireland international, wrote: "We now have a Manager who wants to win games. COYB."
- Sports & Recreation