The Republic of Ireland international has established himself as one of the best right-sided defenders in the Premier League since joining the Toffees for a nominal fee of just £60,000 from Sligo Rovers in 2009.
The Red Devils are known admirers of Coleman, having expressed an interest in bringing him to Old Trafford during the January 2015 transfer window.
United boss Louis van Gaal is looking to bolster his options at the back after being forced to play winger Antonio Valencia out of position on the right defensive flank for much of 2014/15 due to Rafael's failure to impress the Dutchman in his 11 appearances for the club last season.
Van Gaal had been linked with moves for Dani Alves and Nathaniel Clyne, but the former decided to extend his contract at Barcelona, while the latter is expected to leave Southampton for Liverpool in the coming weeks.
However, United are likely to be met with stern resistance from the Merseysiders, who handed Coleman a new five-year deal during the 2014 off-season to secure his long-term future.