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 putting 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.
Albion 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 point 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."
Robin van Persie is fit to face West Brom even though he fell ill on international duty this week but fellow s triker Javier Hernandez is unavailable after he suffered a knee injury in Mexico's friendly against Nigeria
Juan Mata returns after being cup-tied for United's Champions League defeat to Olympiacos
Centre-back Phil Jones (hip) could return but Jonny Evans (calf) and winger Nani (hamstring) are doubtful
Under-pressure Albion head coach Mel reverted to his native Spanish to face the media about Nicolas Anelka's recent ban and speculation over his own future in a tense press conference on Friday
Less than 24 hours before a crucial Premier League home game against United, Mel was forced to bat away questions relating to Anelka's five-match ban for his controversial 'quenelle' gesture as well as his own job security and reports of player unrest.
Albion take on United at The Hawthorns knowing defeat could see them drop into the bottom three and deeper into the relegation mire.
However, the match itself is not top of the news agenda, as Mel referred all questions about Anelka to the club while insisting he is confident about his own future and downplaying talk of dissent among the playing personnel.
The Spaniard has recently used English in press conferences but opted to use an interpreter while under such scrutiny.
"As I can imagine the questions, today I prefer to speak in Spanish," he said.
Asked about his position considering his failure to win in six matches since replacing Steve Clarke, Mel said: "My job depends on winning games and being a head coach is a risky business.
"I came here extremely excited to do a good job and I'm certain that I'm going to carry this out."
Following reports of 'showdown' meetings with players, the former Real Betis boss was eager to stress such conversations are simply routine.
"We've been here six weeks and had six meetings," he said
"There was one this morning but it's intelligent to try and adapt to the style of play that we want.
"They are small things but they will make us better because the players will feel more comfortable
The players are the important ones, that are the coach's bread and butter.
"I'm the third coach at the club this season and it's not been a pleasant situation this season
Winning games changes everything and we could all be here on Saturday evening smiling.
"In football you should never look to the following game and we are totally convinced we can win against Manchester United."
Albion look likely to be without key midfielders Chris Brunt and James Morrison.
Captain Brunt and fellow senior squad member Morrison both returned with knocks from international duty for Northern Ireland and Scotland respectively and will be assessed.
Anelka will begin his five-match ban while right-back Billy Jones (torn hamstring) is out, but Stephane Sessegnon and Claudio Yacob are both back in contention.
Source : PA
Source: PA