Cleverley hopes United can show they have sufficient quality in their ranks to cover for the loss of Fletcher.
"I'm afraid it's bad news for him and bad news for the team," Cleverly said on Sky Sports News.
"He's a massive player for us and it's a shame because we're having a bit of a problem in centre midfield but we've got Giggsy (Ryan Giggs), Ji (Park Ji-sung), Carras (Michael Carrick), Phil Jones doing a job, and Darron Gibson still in the squad, so that's what squads are for.
"He's a fantastic pro, Fletch. He's someone I've always looked up to."