Jose Mourinho has tracked Wayne Rooney for much of the summer, with two bids rejected, but has now moved for the 32-year-old after abandoning his pursuit of the Manchester United man.
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"I am very happy to be here but also anxious because I want to get playing as soon as possible," he told the club's official website.
"It wasn't a hard decision. I saw the qualities Chelsea have, and I was very happy with Jose Mourinho before, so when the opportunity came, I was very happy to take it."
Eto'o becomes Chelsea's second summer recruit from the Russia outfit after Willian's arrival on Wednesday for £31.7m, joining Cristian Cuevas, Andre Schurrle, Mark Schwarzer and Marco van Ginkel at Stamford Bridge.
Goal understands Eto'o's arrival at Chelsea is likely to herald the departure of Ba on a season-long loan. The Senegal striker has attracted strong interest from West Ham and Roma.