By Friday afternoon 17,441 people had signed the online petition calling for Cleverley to be excluded from England's World Cup squad.
The petition was set up by Glenn McConnell from Blackpool, who criticised the player's "inept" performances for Manchester United and England.
Three Lions boss Roy Hodgson was appalled by the petition, under which several fans of the national side have left derogatory comments about the 23-year-old.
And on Friday Moyes sprang to the defence of the midfielder, who has 13 England caps.
"Tom Cleverley is a fantastic young player who's developing all the time," the Manchester United manager said.
"He is a great trainer. He is first out and every day he does his best in training.
"He always gives his best and that is why he will succeed."
Hodgson was unhappy that Cleverley - someone he described as a "sensitive young man" - would have to deal with the pressure of knowing such a petition was out there in the public domain.
But criticism of top-flight players is nothing new, as Moyes acknowledged.
"I remember when Rio (Ferdinand) came in to the (England) team, and Patrice Evra came into the (France) team and they got criticised," Moyes added.
"It's part and parcel, but I think it's wrong on Tom Cleverley."
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