As reported by Goal, Juventus had made a €20 million (£18m/$23m) offer for the CKSA Moscow man and appeared to be in pole position for his signature but the Blues have since stolen a march on the Serie A champions.
The Blues, who are owned by Russian billionare Roman Abramovich, have assured CSKA that they will get closer to their €30m (£27m/$35m) asking price for their star man.
Chelsea, meanwhile, are still yet to clarify their managerial situation for next season, but feel confident that they will land Maurizio Sarri and want to provide the ex-Napoli boss with new additions to the playing staff.
The Premier League side have accelerated their transfer plans this week after direct talks with Sarri were initiated that would see him replace Antonio Conte.
Former Juventus boss Conte remains the head coach at Stamford Bridge, but he will soon be moved aside with his relationship with several players, along with the club's director Marina Granovskaia, having turned sour.
The strength of Chelsea's approach for Golovin has led to Juventus to believe that their chances of landing the Russia star are over, with Golovin, and CSKA, now seeing the Serie A giants as a back-up option.
Lazio midfielder Sergej Milinkovic-Savic had already been dismissed by the Turin giants after the Biancocelesti demanded a huge fee of €150m (£132m/$175m).
Sarri has already spoken to Chelsea about a number of his personal targets, including Gonzalo Higuain, Dries Mertens and Kalidou Koulibaly, but the Blues will retain control of transfer policy, as they have for all their managers.