London born and with his contract having only a year to run, Sterling is open to the prospect of joining Chelsea this summer.
Both Boehly and manager Thomas Tuchel are keen, with the former seeing the England international as a landmark signing to symbolise the club's new era.
Sterling has already held positive discussions with Tuchel over how he would be utilised at Stamford Bridge, with that conversation solidifying the 27-year-old's desire to leave the Etihad Stadium.
A formal bid is expected in the coming days, with City prepared to sell for £55m. For his part, Sterling is seeking £300,000-a-week to sign with Chelsea.