Sterling has entered the final year of his contract at Manchester City and is keeping his options open ahead of making a decision about his next move.
Sources have told ESPN that Chelsea have emerged as serious contenders to sign the 27-year-old, who has already been impressed by Boehly's ambition after his takeover at Stamford Bridge.
City would prefer to receive a fee for Sterling this summer rather than allowing him to leave for free in 12 months and value the former Liverpool winger at between £50-60 million. Sterling, who is keen to test himself in another of Europe's top five leagues, is also attracting interest from Bayern Munich and Real Madrid.
According to sources, Sterling is acutely aware that his next contract will be the most important of his career, and his reluctance to sign a new deal at the Etihad Stadium has centred on doubts about whether he is central to Pep Guardiola's plans for the future.
Guardiola has made a point during his time at City of not keeping players who are unhappy. Gabriel Jesus, who also has a year left on his contract, is expected to leave to join Arsenal after telling club bosses he would like the opportunity to play more regularly.
If Sterling was to make the same plea City would be unlikely to stand in his way, particularly if it was made clear he does not plan to sign a new contract.
Guardiola is already well covered if both Sterling and Jesus depart this summer.
Striker Erling Haaland has joined from Borussia Dortmund, while Argentine forward Julian Alvarez, signed from River Plate in January, has been told he is set to be part of the first-team squad next season and will not be loaned out.