The Stamford Bridge club are also interested in Kasper Schmeichel and Gianluigi Donnarumma amid expectations that the Belgian will leave west London this summer.
Arsenal are open to selling Cech to provide manager Unai Emery with the funds needed to continue rebuilding his squad.
The veteran goalkeeper has already turned down a move to Napoli to stay at the Emirates and fight for his place with new arrival Bernd Leno.
Leno is one of five first-team arrivals at Arsenal already this summer.
He is understood to have a good relationship with Chelsea director Marina Granovskaia, which could help ease a smooth transition back to the club.
Real have made Courtois their primary goalkeeping target for the window and the Belgian has dropped hints over his uncertain future at Stamford Bridge, which has put the Spanish giants on red alert.
Chelsea are hopeful of concluding a £22m deal for Aleksandr Golovin from CSKA Moscow ahead of Monaco. However, the finer details are still to be agreed.
‘In order not to ruin [Golovin] we must let him go. But we have no solution for today,’ CSKA president Evgeniy Giner told Russian outlet Sport24.
‘When we receive an offer, we always try to negotiate openly. Indeed, we were visited by the vice-president of Monaco [Vadim Vasilyev] and made a proposal for Golovin.
‘Chelsea? There is no closed arrangement with them. But the question of his transition to another club will be decided, I think, in the next couple of days.’