Left-back Cole had resigned himself months ago to a summer move from Stamford Bridge, where he spent eight seasons, and was not offered fresh terms by the club.
Eto'o, 33, scored 12 goals in 35 appearances after signing a one-year deal with the Blues last August, but was not deemed worthy of an extension by Jose Mourinho, who has agreed a deal for Atletico Madrid forward Diego Costa to move to West London for next campaign.
"Cole has a very strong claim to be the club's greatest player of all time in his position," read a Chelsea statement.
"Chelsea thanks Ashley for the fantastic service throughout his eight years here and for helping to bring so many trophies home."
Cole, also 33, won eight major trophies at Chelsea, including the Premier League and Champions League.