The 27-year-old was on the books at Anfield for five years between 2010 and 2015 and emerged as one of the Premier League's brightest prospects under Brendan Rodgers.
Sterling registered 23 goals and 25 assists in 129 senior appearances for Liverpool, but he departed in acrimonious circumstances for Manchester City in a £57.3m deal seven years ago.
Sterling won 12 major trophies at the Etihad before a £47.5m switch to Chelsea was agreed earlier this month, seeing him return to London 12 years after his departure from Queens Park Rangers.
A move to La Liga with Barcelona was previously mooted for Sterling, who was also briefly tipped to reunite with former Man City assistant Mikel Arteta at Arsenal.
However, speaking to GIVEMESPORT, journalist Jacques Talbot claims that Sterling was interested in returning to Anfield, but Liverpool chiefs did not consider a move for the England international.
"We heard from reliable sources that Sterling actually wanted to go to Liverpool before making the move for Chelsea. I don't think Manchester United were a consideration, but what I heard is that in essence, Liverpool rejected him," Talbot explained.
Liverpool have lost Sadio Mane and Takumi Minamino from the flanks this summer, but Fabio Carvalho has arrived from Fulham and Mohamed Salah ended speculation over his future by signing a new three-year deal.
After his exit from Liverpool, Sterling became a four-time Premier League champion with Man City and amassed 131 goals and 94 assists in 337 outings in all tournaments for the club.
The attacker was due to become a free agent in 2023, though, and he expressed his delight at returning to the English capital after linking up with Thomas Tuchel's squad.
"First and foremost, it's a pleasure to be here. I've obviously achieved a lot in my career so far, but there is still so much more to achieve and I'm really looking forward to doing that in a Chelsea shirt, under Thomas's management," Sterling told chelseafc.com.
"London is my home and where it all started for me, and it's amazing I now have the opportunity to play in front of friends and family week in, week out at Stamford Bridge. I'm really looking forward to meeting the fans there soon.
"I do want to take the opportunity to thank Todd, Behdad, the ownership group, Thomas, and all involved in the process of getting me here. I can't wait to get going now and continue to do my talking on the pitch."
Chelsea kicked off their pre-season with a 2-1 win over Club America in Nevada on Saturday evening, but neither Sterling nor Kalidou Koulibaly made their debuts for the Blues in that game.
The Blues will now prepare to face off against Charlotte and Arsenal in the United States before travelling to Italy to meet Udinese in their final pre-season friendly on July 29.