The 22-year-old has been attracting interest from Europe’s leading clubs for some time, with his talents first noted at the Confederations Cup in 2017.
He enhanced that reputation by starring for hosts Russia at the 2018 World Cup, with a big-money summer transfer now expected.
Monaco, though, have sought to surge to the front of the queue by making a formal approach, with the Ligue 1 outfit of the opinion that they could best serve the ongoing development of an exciting young talent.
The club’s vice-president, Vadim Vasilyev, told Sport Express: "We made CSKA a proposal for Golovin.
“I cannot reveal the amount of the transfer, but it is a worthy fee. The duration of the contract? Standard, five years.
“I think in this situation the transition to Monaco is the most correct option for Golovin.
“To go from Russia immediately to the top club of the leading European league is very risky. We are able to launch the career of young players – just look at Kylian Mbappe.
“Everything should be decided in the coming days. Probably, even today.
“But everyone needs to understand that we cannot wait long. If the transaction does not succeed, we will move on to the next target.”
Golovin has emerged out of CSKA Moscow’s academy system to make 113 senior appearances for the club.
The 2017-18 campaign proved to be his most productive to date, as he netted seven times, before an international cap tally was taken to 23 at a home World Cup.