Golovin, 22, attracted interest from LaLiga giants FC Barcelona before the start of the global showpiece, but the interest soon died down with the Spanish side unconvinced he'd settle into life at the Camp Nou.
Chelsea moved swiftly to agree on a reported £27m (R491m) transfer but CSKA President Evgeniy Giner has revealed a host of sides have since entered the race - including Ligue 1 giants AS Monaco.
"I believe that [he] can grow into a big star," Giner told Russian outlet sport24.
"I'm not sure that this can happen in the Russian championship. The level of the English Premier League or the championship of Spain is much higher.
"In order not to ruin [him], we must let him go," he added in reference to the development of the tireless playmaker.
"But we have no solution for today. When we receive an offer, we always try to negotiate openly. Indeed, we were visited by the vice-president of Monaco [Vadim Vasilyev] and he made a proposal for Golovin.
"If we talk about Chelsea, then there is no closed agreement with them. But the question of his transition will be decided, I think, in the next couple of days."