According to reports, both clubs are interested in signing the 20-year-old on a season-long loan deal.
The midfielder spent last season on loan at Championship side Watford where he impressed, scoring 11 goals during his time at Vicarage Road.
Cleverley joined United as a trainee at the age of 15 and is expected to sign a new contract to keep him at Old Trafford.
United boss Sir Alex Ferguson is said to rate the youngster highly and has no intention of letting him leave on a permanent basis.
However, Ferguson could be prepared to allow Cleverley to leave on loan again in order to gain valuable Premier League experience.