The message, "Wrong One - Moyes Out," is in reaction to a controversial "Chosen One" banner that hangs inside the stadium, with sections of supporters keen to vent their frustration at the Scot.
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The former Everton boss, though, was quick to downplay the significance of the protest, which will take place during Saturday's game against Aston Villa, and hinted that setbacks were to be expected as he begins rebuilding the club.
"I have heard of it and what they are saying about it, but this is a long journey," said Moyes when asked about the banner at his press conference on Friday. "This is only the start of the journey. People are entitled to do that."
Protest organizer Wes Jones, meanwhile, says the purpose of the "Wrong One" campaign is to highlight that United fans are not totally united behind Moyes.
"The rumblings of discontent have started in the stands," Jones told the BBC. "We wanted to show that support isn't as 100 percent as Moyes would like to think."