The Manchester United man was included Roy Hodgson's 30-man party for Wednesday's 1-0 friendly victory over Denmark at Wembley, and was named as a matchday substitute.
Cleverley did not make an appearance in the fixture, but was booed by sections of the Wembley crowd when his name was read out over the public address system before kick-off.
The petition - which has garnered over 17,000 signatures thus far - states that Cleverley has "been regularly selected by Roy Hodgson in his squads without possessing any genuine qualities whatsoever".
Hodgson himself condemned the petition, and insisted the 24-year-old has "done an awful lot of good" for England.
United manager Moyes has now also leapt to the defence of his player, and says it is "wrong" that the social media campaign has been given so much credence.
"First and foremost I am surprised that the media would be picking something up on social media and making a big case out of it because that is wrong," he said.
"Tom Cleverley is a fantastic young player who is developing all the time. He is a great trainer and he is someone who is first out and every day does his best in training."
Cleverley has made 13 international appearances since making his debut against Italy in August 2012.
