Over 17,000 people have signed the petition, which was set up by Glenn McConnell from Blackpool, who criticised the player's "inept" performances for United and England.
"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."
Cleverley received the backing of England manager Roy Hodgson, who lashed out at those who had signed the petition.
The midfielder did not feature for England in their friendly win over Denmark but he is fit for United's game against West Brom on Saturday.
A letter from Moyes to United season ticket holders came to light on Friday morning.
In it, the Scot thanked the fans for their support during what he conceded was a sub-par season.
Moyes repeated that message at his pre-match press conference.
"We have great support," he said.
"The supporters understand the difficult period we are in and they are backing the players.
"I believe that the players are working hard for each other. They know they can play better, and they will play better."
Last week's Capital One Cup final and the international break mean Moyes' team have been without a match for 11 days.
Their last performance was one of the worst of the season. A 2-0 defeat against Olympiacos put Moyes' men on the brink of elimination from the Champions League.
The big gap between fixtures has been both a help and hindrance to the United boss.
"It has allowed some players to get games, like Darren Fletcher, Shinji Kagawa and Chicharito (Javier Hernandez), the people who have not featured as much as we would like," Moyes said.
"But certainly it's also been a hindrance because we went into it off a bad performance and result in Olympiacos and we wanted to put it right much sooner."
United have the advantage of facing a team in a far worse predicament than their own this weekend.
West Brom have not won a match since Pepe Mel replaced Steve Clarke as head coach of the midlands club.
West Brom were forced to deny speculation that Mel was on the verge of the sack despite only being in the job less than two months.
Moyes is sure the Spaniard's job is not under threat even though the Baggies are just one points above the drop zone.
"I'd be very surprised if Pepe Mel is under pressure at this moment," he said.
"I wouldn't think for one minute that after six games they were (thinking about) bringing in a new manager."
Clarke led West Brom to eighth in the league last season, but he was axed in December following a poor run of form.
Moyes admits he was surprised to see his compatriot sacked.
"I'm not privy to what goes on at West Brom but I thought he did a great job last year and had them in a lofty position in the Premier League," Moyes added.
"When I was at Everton last year, they were challenging us a lot of the time and he did really well so I'm surprised Steve left.
"I'm not aware what has gone on at West Brom but he's a good manager and proved that in his time at West Brom."