Moyes privately disliked the banner, displayed by the much criticised Stretford End Flags group back in August, and admitted he was not even deserving of it.
Supporters in the second tier of United's Stretford End attempted to rip the banner down during March's 3-0 derby defeat to Manchester City, as stewards were tasked with protecting it.
Four days later, a set of supporters arranged for a flyover during the Aston Villa match which read "Wrong One - Moyes Out," as they showed their contempt for the banner and United's decision to appoint Moyes.
A spokesman for the National Football Museum in Manchester appeared to dash the possibility of exhibiting the banner.
"While we are always grateful for offers, we often have to decline for reasons of space and because it doesn't fit within our guidelines," they said.