Chelsea are seeking two attacking midfield signings this month.
Pulisic's is the USA's most-talented player in a generation and has made waves at Borussia Dortmund.
CSKA Moscow playmaker Golovin, meanwhile, has caught the eye while playing for Russia at the World Cup.
Chelsea are under pressure to put this season's disappointing fifth-place finish behind them next term.
Pulisic and Golovin will be seen as players that can help kickstart a promising new era for the Blues, who are in for a "busy few weeks" before the transfer window closes next month.