According to ESPN, the Blues are lining up the Hungary international as a potential replacement for Thibaut Courtois, who is yet to renew his contract at Stamford Bridge.
Antonio Conte's side are said to face steep competition for Gulacsi's signature, with Real Madrid and Paris Saint-Germain also linked with the 27-year-old.
The Hungarian has been touted as one Europe's top goalkeepers, and is reportedly rated at around £25m.
Gulacsi, a product of Liverpool's academy, joined the German outfit from affiliate club RB Salzburg in 2015 and is currently under contract until 2022.
The five years he spent with Liverpool's youth setup would see him qualify as a homegrown player should he sign for a Premier League club.