The 22-year-old played for new Blades boss Ruben Selles on-loan with Hull City last season.
Barry told the club's website, "The first time I had conversations about it, I just couldn't wait to come here. The season is three weeks away, and I know we have a couple of friendlies before, but I am just buzzing to get started.
"I've been at Villa, playing games, training and working hard just for whatever will come next. As I say, when a club like this comes calling, you can't turn it down and I can't wait to get into it.
"I love his (Rubén Sellés) style of play, it is exactly what I stand for as a player. I just want to get to the goal as quick as possible. I have got a really good relationship with the gaffer and the staff, so I can't wait to get into training now."