After a successful loan period with the Blades last season, Cotterill has joined on a permenant basis for an undisclosed fee believed to be in the region of £1-2 million.
Speaking to the official website, Steve Bruce said: "Cotts is clearly a talented player who could have a big future in the game.
"He became a full international while still a teenager and he feels he would be better suited enjoying regular first team football at this stage of his career.
"Sheffield have made us a good offer, one we feel is too good to turn down. We wish him all the best in the future."
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