This is according to Sky Sports sources, who understand Chelsea are closely monitoring the England international's situation with a view to making a formal offer.
Sterling wants a move back to London and would prefer to join Arsenal, but is not against the idea of joining Manchester City, who are expected to table an improved bid after their initial £25 million offer was rejected by Liverpool last week.
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Man City could increase their opening offer by as much as £15 million as they attempt to secure the 20-year-old's signature as soon as possible.
But news that Chelsea have entered the race for his services may have alerted Sterling, who is keen to join a club where he feels his chances of winning silverware are enhanced.
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Sterling wants to leave Liverpool
The former Queens Park Rangers starlet, who joined Liverpool in 2010 before making his first-team debut in 2012, has made it clear to the Merseyside outfit that he has no plans to renew his contract, which expires in the summer of 2017.
He insists that it's not a case of wanting more money - he rejected Liverpool's most-recent £100,000-a-week offer - even though it's expected he will earn a more lucrative deal away from Anfield.
Reasons why Mourinho wants Sterling
There are several reasons why Chelsea boss Jose Mourinho will be interested in signing the youngster.
First and foremost, Sterling is an exciting winger with an extremely bright future ahead of him. Last year he picked up the coveted Golden Boy award - an accolade previously won by the likes of Lionel Messi, Cesc Fabregas and Sergio Aguero - and has shown he is blessed with immense talent over the past two seasons.
Second, neither Willian or Juan Cuadrado have managed to asset themselves on the right wing in the same way Eden Hazard has on the opposite flank. In fact, there are already rumours that Cuadrado could leave Stamford Bridge this summer.
Sterling plays his best football from wide-right, but can also play through the middle, as he demonstrated on occasions during the 2014-15 campaign with Liverpool.
And last, but by no means least, Mourinho will not be content to sit back and watch his great adversary Arsene Wenger take Sterling to the Emirates Stadium without a fight.
Chelsea, as the reigning Premier League champions with arguably a better chance of winning the Champions League than Arsenal within the forthcoming years, could prove to be an even more attractive option for the unsettled Sterling.
Chelsea fans: would Raheem Sterling be an upgrade on Juan Cuadrado and/or Willian? Have your say by leaving a comment in the box below!