The 27-year-old was again an important player for the Citizens during the 2021-22 campaign, making 47 appearances in all competitions, contributing 17 goals and nine assists in the process.
Sterling's contract at Etihad Stadium is due to expire next June, though, and there has recently been speculation surrounding his future in Manchester.
According to journalist Ekrem Konur, Los Blancos boss Ancelotti has now approved the signing of Sterling during this summer's transfer window.
The attacker scored 13 goals and registered six assists in 30 Premier League appearances last term, while he contributed three goals and two assists in 12 Champions League outings.
Sterling was not a starter in the latter stages of Man City's 2021-22 European campaign, though, and he was only a second-half substitute against Aston Villa on the final weekend of the Premier League season.
Man City have already boosted their attack this summer with the signing of Erling Braut Haaland, while Jack Grealish will be looking to improve after an underwhelming first season with the Citizens, which could further limit Sterling's playing time at the club.
Real Madrid have already completed the signings of Antonio Rudiger and Aurelien Tchouameni, with the former arriving on a free transfer from Chelsea, while the latter has joined in a big-money deal from Monaco.
However, the club were prepared to commit to a huge deal for Kylian Mbappe before the France international decided to sign a new contract at Paris Saint-Germain.
Vinicius Junior, Karim Benzema, Rodrygo and Eden Hazard are all staying at the club this summer, but Ancelotti's attacking options could be weakened by the possible departures of Marco Asensio, Mariano Diaz and Luka Jovic.