The action is in response to the contentious 'Chosen One' banner that is on display at Old Trafford despite calls for it to be removed in the wake of United's increasingly troubled season under Moyes.
A United fans' blog, therepublikofmancunia.com, claimed supporters are planning to send their own message during the game against Aston Villa on Saturday.
The blog states: "A group of United fans have raised £840 in a day to fund a plane to fly over Old Trafford with a message reading 'Wrong One - Moyes Out'.
"The message is a response to the Stretford End Flags banner which reads 'The Chosen One'."
United would not be drawn on the threat, but there is no doubt the 'Chosen One' banner is causing controversy.
After United stumbled to a 3-0 defeat at home to Manchester City, there were angry protests directed against Moyes and his predecessor Sir Alex Ferguson.
Some fans had earlier called for the group that made the banner - Stretford End Flags (SEF) - to remove it following the 3-0 defeat to Liverpool almost two weeks ago.
Stewards also gathered around the banner at full-time after the City defeat to prevent it from being seized.