McGahey has put pen to paper on a two-year deal with the option of another 12 months and becomes the club's ninth summer signing.
The transfer fee for the 18-year-old will be decided by a Football League tribunal after the two clubs failed to reach agreement.
Blades boss Nigel Clough told the club's official website: "A number of the coaching staff watched Harrison at the end of last season and he has potential, that's what we see in him.
"He will admit that he is not the finished article but we are delighted that he has chosen to join us to aid his development.
"Harrison is a big, strong defender and there are a number of similar characters at Bramall Lane, from whom he can learn his trade."
McGahey made four appearances for Blackpool last season and has had spells at Blackburn and Liverpool.