The England international – who missed out on Roy Hodgson’s World Cup squad – is due to enter the final 12 months of his deal in the coming weeks and was set for talks to extend his stay at Old Trafford.
But after discussions with incoming manager Louis van Gaal is understood that any plans to thrash out a new contract will be shelved until later this summer.
Cleverley is thought to want assurances over his role with United being linked with a number of midfielders including the likes of Bastian Schweinsteiger from Bayern Munich.
Everton and Arsenal are believed to have been alerted by the situation, and could be ready to offer the playmaker a fresh start.
The Toffees have been linked with Cleverley on a number of occasions in recent months, with Roberto Martinez considered to be a big fan of the 24-year-old.
Arsenal are also keen, with a £6m offer having been mooted in the past.