Patrick Roberts is highly regarded amongst coaches
The deal for Roberts makes him the third English player City have signed in a week after the transfers of Raheem Sterling and Fabian Delph.
City have targeted English players in the transfer market this summer, in a attempt to comply with the homegrown rule and to counter the departures of James Milner, Frank Lampard, Dedryck Boyata, Scott Sinclair and Micah Richards.
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The 18-year-old is highly rated and has represented England at Under-16, Under-17 and Under-19 levels.
He made 22 first-team appearances during his time with Cottagers and played a key role at they reached the 2014 FA Youth Cup final.