The 27-year-old has 12 months remaining on his current United deal, but is yet to be offered fresh terms with the club willing to listen to offers for the long-serving Northern Irishman.
United are understood to value Evans at £10million, with Everton feeling the player’s value lies at nearer £6million.
But the Daily Telegraph claims a compromise deal of £8million is on the verge of being struck between the two clubs.
Senior figures at Old Trafford have refused to comment on Everton’s interest in Evans, but it is understood that Toffees boss Roberto Martinez sees the player as the ideal replacement for Sylvan Distin, who moved to Premier League new boys Bournemouth earlier this summer.
United manager Louis van Gaal is prepared to keep Evans in his squad, but has made it clear to the defender that he has dropped behind Chris Smalling, Phil Jones and Marcos Rojo in the pecking order at Old Trafford.
And with United also in the market for a world-class centre-half, with Real Madrid’s Sergio Ramos van Gaal’s number one target, Evans is facing up to the prospect of diminished opportunities if he stays at the club.
Evans, currently in Seattle with the United squad on the first leg of their pre-season tour of the United States, has made 198 senior appearances for the club since making his time in a League Cup tie against Coventry in September 2007.
Never really more than a squad player, Evans has nonetheless won three Premier League title medals during his eight seasons at Old Trafford.