One of the World Cup's standout stars is said to have been made the subject of a £17.7m bid by Juve earlier this window, while new Chelsea boss Maurizio Sarri is also a fan and intends to make a move of his own.
Monaco now appear to have moved to the front of the queue, however, with the French club's vice-president Vadim Vasilyev hoping for a swift response from CSKA.
"[It's] a more than generous offer," Sky Sports News quotes him as saying. "The contract length offered is five years. I think that a club in transition like Monaco is the best option for him, at this stage in his career.
"Everything should be decided in the coming days, maybe even today. We cannot wait long. If it does not come off, we will look to another option."
Golovin, who scored one and set up two more for World Cup quarter-finalists Russia, is under contract at the VEB Arena until 2021.