The powerful 19-year-old striker, Blades manager Danny Wilson's first summer signing, will stay at Bramall Lane initially until January with a view to a season-long loan.
Wilson said: "We're very short in that area, that's for sure and he'll give us pace and power. He's got a great attitude as you'd expect from somebody who's played under Sir Alex Ferguson and he's got a great talent.
"He's one of the young ones again that they seem to churn out all the time, these quality players, but they need a place to play and we were fortunate that they allowed us to bring him here and see how he goes on.
"There's no pressure on the boy whatsoever, he's still a young lad, but what we were missing was that pace and power."
Cofie was born in Ghana, but has represented England at youth level and moved to Old Trafford in the summer of 2009 after starting out at Burnley. He spent last season on loan at Belgian side Royal Antwerp.