The 27-year-old Jamaican-born star has one year remaining on his contract at Manchester City.
But latest reports have claimed that, would take a bid close to £60m from Chelsea for a deal to be done in this transfer window.
Chelsea manager Thomas Tuchel, is also understood to be a big admirer of Sterling, with the Blues boss believing the attacker is better suited to Chelsea’s style of play than Romelu Lukaku – who looks set for a loan return back to Italy with talks progressing with Inter Milan.
According to Sky Italy, Chelsea believe a fairer price would be around £35m for the Englishman’s services.
The 27-year-old attacker has played 339 times for the Premier league champions scoring 131 goals – just outside the top ten in the club’s all-time goalscorers list.
However, a guaranteed starting spot hasn’t always been provided for him under manager Pep Guardiola.