The 27-year-old has now entered the final 12 months of his contract with the Premier League champions, who are desperate to avoid losing him on a free transfer in 2023.
Sterling is understood to have been representing himself in contract negotiations with City, which are yet to bear fruit, and he is edging closer to an exit from the Etihad.
According to the Daily Mail, Chelsea and Man City have now made major progress in negotiations over a deal, and there is hope on both sides that a move can be concluded with few difficulties.
The report adds that Chelsea will fork out £45m to sign Sterling, who will sign a four-year contract at Stamford Bridge until the summer of 2026, and a deal could be finalised by the end of the week.
Sterling therefore prepares to leave Man City after a trophy-laden seven years since his £57.3m arrival from Liverpool, during which he has scored 131 goals and provided 94 assists in 337 games in all competitions.
The England international has won four Premier League titles, one FA Cup, five EFL Cups and two Community Shields with Pep Guardiola's side and enjoyed a productive campaign on a personal level last season.
Sterling registered 17 goals and nine assists in 47 appearances for the Citizens but was not a regular fixture of Guardiola's XI, starting 23 Premier League games and being an unused substitute in seven of them.
Guardiola's ranks out wide have already been thinned by the departure of Gabriel Jesus to Arsenal in a £45m deal, although the transfer is yet to be confirmed by either club.
Riyad Mahrez's long-term future has also been called into question ahead of the expiration of his contract in 2023, although the 31-year-old reportedly wants to stay and has been offered a new deal until 2025.
However, City may be reluctant to let more than one player depart to a direct rival, potentially forcing Chelsea to look elsewhere in their search for defensive reinforcements.
Marcos Alonso and Cesar Azpilicueta could both follow Antonio Rudiger and Andreas Christensen out of the exit door in the coming weeks, and Juventus centre-back Matthijs de Ligt remains the Blues' top centre-back target.