Several Newcastle players, including Steven Taylor and former Toon defender Danny Simpson, cancelled their Twitter accounts because of abuse.
One former Newcastle player told us: "The world is full of wankers who hide behind their keyboards and give us abuse - ALWAYS hiding their email addresses.
"They aren't supporters, they aren't even football fans, most of them, just nurds wanting to ruin someone's day.
"Is Twitter and Facebook worth the effort? I don't think it is."
The petition, which accuses Cleverley of "inept" displays, has gained over 12,500 signatures since it was posted.
Hodgson, who has included the 24-year-old in Wednesday's squad to face Denmark, said: "It doesn't impress me.
"He deserves to be in this 30-man group. He's done an awful lot of good.
"I don't think it's correct people should single out one player for Manchester United's lack of success this season.
"I would like to think that I'm not going to have the England team selected in the future by petitions from various people who are holding it against one person for a team's lack of success.
"You can't expect a football coach to have any sympathy with it.
"Whether he'll make the 23 or not will of course depend on a lot of things, not least of all how he does in the next couple of months.
"But that doesn't just apply to Tom Cleverley, that applies to a good many players. I'm disappointed that Tom, as a sensitive young man, has got to bear that burden.
"But I know one thing... he will come through it and when he does come through it and Manchester United start winning again and he's starting to get praise for his performances, he'll be stronger for it."
The petition was started on the online campaigning platform Change.org by Glenn McConnell of Blackpool.
Former England midfielder Darren Anderton: "It's scandalous!
"You're talking about a young lad who wants to play football for his country and what a ridiculous thing that petition is. There's no need for it.
"Maybe it's an anti-Manchester United thing, but there's absolutely no need for it. The manager has lots of good players to pick from and if he wants to pick Cleverley then so be it and hopefully he performs very well."