The Blues have been looking to add a goalkeeper to their squad following the sale of Edouard Mendy to Al-Ahli in Saudi Arabia.
Sanchez, 25, will challenge Kepa Arrizabalaga for the number one spot.
The Spaniard will be familiar to the coaching staff at Chelsea as goalkeeping coach Ben Roberts worked with him while at Brighton.
Sanchez has been with the Seagulls since the age of 15, having joined their academy from Levante in 2013.
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