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Thomas Tuchel is a big admirer of Sterling, with the Blues boss believing the forward is better suited to Chelsea's style of play than Romelu Lukaku, who is set to go out on loan back to former club Inter Milan next season.
However, City are understood to value Sterling at up to £60m, even though the player only has one year left on his contract at the Etihad.
Chelsea, meanwhile, believe a fairer price would be closer to the £35m which Bayern Munich could end up paying for Liverpool forward Sadio Mane.
Mane also has just 12 months left on his deal at Anfield, but at 30 is two years and eight months older than Sterling.
Sky in Italy have reported a £35m offer from Chelsea for Sterling.
Sterling has won four Premier League titles, four League Cups and the FA Cup at City, after joining the club in the summer of 2015.
The 27-year-old has played 339 times for the current top-flight champions, scoring 131 goals - just outside the top 10 in the club's all-time goalscorers list.
But a guaranteed starting spot has not always been provided for Sterling under Pep Guardiola. The England winger started just five out of a possible 12 Champions League games this season and was unused by City's Spanish manager nine times in the Premier League last term.
Sterling started his career at Chelsea's west London rivals Queens Park Rangers, before moving to Liverpool when he was 15.
Sterling has family connections in London
Sky Sports News reporter Mark McAdam:
"Chelsea fans have been getting worried about a lack of transfer activity compared to their rivals - Todd Boehly and the new Chelsea ownership could be making a huge statement here.
"Raheem Sterling wants to play football. He wants to be a part of things. He's been a huge part of what Manchester City have done, winning four Premier League titles, four League Cups and the FA Cup and a Champions League runner-up medal as well. He's been a huge part of their success story, as well as 77 England caps.
"We all know the World Cup is coming up in November and December and while Sterling has not been a huge part of City's starting XI in the last season, it's that guaranteed football that is going to be of interest to Gareth Southgate and Sterling.
"There's a nice story in here as well. We know Sterling started his career in the Queen's Park Rangers academy in west London. He grew up in the shadow of the Wembley Stadium arch and spent a lot of time in London before he went up to Liverpool.
"He has that family connection and history down in London. That will be a big factor in this situation. Chelsea are big players in the Premier League and have won lots of trophies. That's what Sterling wants to do: Play at the highest level. He wouldn't just be coming home, but he'd be part of a successful team under new owners, who want to carry on winning things. This is going to be a fascinating story that will play out over the summer, but the first moves have been happening today."
Sterling to Chelsea? Your views...
Jetsdad: Absolutely madness if City sold him to potential title rivals it could really bite them in the backside. Why is he so out of favour?
Pierre: I'd rather take Dembele for free and not Sterling even if he's a proven player and could do a deal. Our transfer budget should prioritise defence
Jack lowe: Sterling to Chelsea? Good deal, 27 experienced winger. Scores goals. But not for 60 million. In that case I'd rather Dembele who is younger and would cost very little compared to sterling
Benjy: As a Chelsea Fan, for me sterling is a no-brainer, there not many prem proven ayers around who are available, he guarantees a healthy amount of goals and a more reliable option than our current crop off attackers.
Ollie: Makes sense all round, no club would turn down a fit and firing Sterling and Sterling needs games to cement his place in the WC starting XI with all the young talent coming up behind him.
Callum: It would be a big mistake, City selling Sterling to Chelsea. It weakens them and strengthens one of their rivals. I'd have him back at Liverpool to replace mane but I think most fans wouldn't accept him. He left on very bad terms but you can imagine what Jurgen Klopp could do.
Andy: Premier League proven, a leader and more consistent finisher than what Chelsea have now, Providing the fee is below £50m its worth it but anything more is another expensive gamble
Follow the summer transfer window with Sky Sports
Who will be on the move this summer before the transfer window closes at 11pm on September 1?
Keep up-to-date with all the latest transfer news and rumours in our dedicated Transfer Centre blog on Sky Sports' digital platforms. You can also catch up with the ins, outs and analysis on Sky Sports News.