Pep Guardiola confirmed this week that City would only look to sign one or two players during the summer transfer window and said the club were not in a position to spend another £300m on signings.
A winger who can also operate through the middle is believed to be a priority for the City manager, who committed his future to the club this week by signing a new deal.
Hazard, who has won the Premier League twice at Chelsea, fits the bill and has suggested he could leave if the Blues do not make the right signings in the summer.
The Belgium superstar is believed to have a strained relationship with Antonio Conte and could look to move even if the Italian departs as expected.
Real Madrid have been long-term admirers of Hazard but The Sun say Manchester City will rival the Spanish giants for his signature this summer.
It is claimed the new Premier League champions would not hesitate to launch a £100m-plus bid if given an indication that a deal could be possible.
City would also have no issue giving Hazard, a player Guardiola was reportedly first attracted to at Bayern Munich, a huge pay rise.
Discussing his plans for the summer transfer window this week, Guardiola told Sky Sports: Yaya [Toure] is leaving so we have to replace him, and maybe another one.
One or two more, no more. We cannot invest…maybe people dont believe me but we dont have the money to invest £300m every season.