The Russia international has been a long-term target for the Blues.
The Sun says Blues owner Roman Abramovich is willing to pay around £15million for his fellow Russian.
He could either bring him into the squad for Antonio Conte in January or send him back to his homeland on loan for the rest of the season.
Golovin, 21, is tipped to be the hosts' leading light at the World Cup and Chelsea want to tie him down now rather than face an auction if he does well.
However, Abramovich is also conscious of the fact that Golovin needs to play in the second half of the season, so he could be bought and then left in Russian club football.