The midfielder recently helped the Blues clinch their second title within three years, but despite playing over 150 times for the club it is thought the 28-year-old wants more starting appearances.
His 31 games for the Blues in their 2013-14 title-winning campaign included 12 starts, but with a year left on his current contract there has also been speculation that the club could offer him a new deal this summer.
Everton, Arsenal and Tottenham are said to be heading the list of potential destinations for the England international.
A return to Newcastle, where the Yorkshireman spent four years after his departure from Leeds United in 2004, has also been mooted.
Milner recently admitted that whether he would be better suited by a move away from Etihad Stadium was in his thinking.
He said: "It's an ongoing question whether I am better staying off at a club like City and playing big games in big competitions or playing more regularly somewhere else."