The Mirror claims that the 23-year-old Englishman will be recruited as an understudy to first-choice custodian Petr Cech.
Thibaut Courtois is scheduled to return to Stamford Bridge from his third successive season on loan at Atletico Madrid in the summer, but the Belgian could be sent back again.
Blues manager Jose Mourinho has instructed his scouts to find three or four home-grown players who would be valuable squad players for next season.
Darlow, who has been at Forest his entire career, has made 39 appearances in the Championship so far this season, making 57 saves and keeping 11 clean sheets in the process.
Darlow's current contract with Forest runs out in June 2017.