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The 27-year-old has one year remaining on his City contract and there's a 'growing feeling' among potential suitors he'll be ready to leave.
Per the Telegraph, a number of top European sides are said to be interested in Sterling including Spanish giants Barcelona and Real Madrid. Bayern Munich are monitoring the situation in case a move for Sadio Mane breaks down.
However, Chelsea have also emerged as shock contenders for the England international's signature.
The Blues have recently been taken over by an American consortium led by Todd Boehly and Sterling would be a marquee signing given his profile.
Sterling started 23 times for City in their title winning campaign but isn't the regular he once was, with omission from the starting XI of both Champions League semi-final clashes with Real Madrid most noticeable.
He's scored double figures in his last five Premier League seasons and it's believed he'd be attracted to a move to London, where he grew up.
Back in October, Sterling revealed he'd be up for moving abroad if the opportunity presented itself.
He explained during an appearance at the FT Business of Sport US summit: "If there was an opportunity to go somewhere else, (for more game time), I would be open to it at this moment in time.
"As I said, football is the most important thing to me - challenges that I have set myself from a young age and dreams as well, to play abroad.
"As an English player all I know is the Premier League and I have always had something down (in me) that maybe one day I would love to play abroad to see how I would come up against that challenge."
City would still look for around £50 million to let Sterling go, what they paid to sign him from Liverpool in 2015.
Chelsea signed Romelu Lukaku last summer to lead their attack but the transfer has been underwhelming.
Manager Thomas Tuchel isn't against sanctioning a return to Inter Milan, but would want a replacement too.
The reigning world champions finished third in the Premier League but know more goals are required to challenge City and Liverpool at the top.
Lukaku was their top scorer with 15 in all competitions, but it's a below par return for a striker who cost a club record £97.5 million.
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