The 19-year old striker will spend next season on loan with the Blades with manager Danny Wilson hoping he can bring some of his Old Trafford youth academy form with him to Bramall Lane.
"John is another exciting prospect out of the Manchester United Academy," he told the club's official website.
"He has the pace and the ability to cause defences all sorts of problems and most importantly he scores goals.
"We are delighted to have completed this deal as we think it will be very beneficial to all parties."
This will be Cofie's second loan spell away from Manchester following his move to Belgian outfit Royal Antwerp last season.