The England international has been an infrequent starter for the Sky Blues this season, with Manuel Pellegrini having often overlooked him in favour of David Silva, Jesus Navas and Samir Nasri.
This has alerted the Toffees, who are keen to offer the 28-year-old an escape route.
Milner – who has made 18 of his of his 29 Premier League outings from the bench this term – is said to be looking to assess his options this summer, which could open up the possibility of a move.
Everton already have Gareth Barry on loan from the Etihad Stadium, but with the midfielder now entering the back end of his career it remains unclear as to whether or not Roberto Martinez will push to keep him.
Milner is seen as an ideal replacement for the veteran, with the former Leeds youth star able to play effectively in central areas as well as on the wing.
City are likely to want a sizable free for the 28-year-old, who has two years remaining on his contract and cost £26m in 2010.
Tottenham are also interested, but the player himself is settled in the north west.