The England international is known to have been made available by Louis van Gaal, and he has attracted bids.
Both Aston Villa and Valencia have made offers for a permanent deal, but Cleverley is understood to be holding out for Everton.
Everton is a confirmed admirer of Cleverley, although the club refused to let him be a make-weight in any possible deal for James McCarthy.
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Now with Cleverley possibly available on loan, Everton has opened talks and is hoping to clinch a late deal for him.
United could be set to let up to six players leave the club on Monday and Cleverley could very well be one of those.