Davies moves from Stoke City on an initial six-month contract to replace the departed Robin Olsen, although there is an option to extend his stay in South Yorkshire beyond the summer.
“I'm absolutely buzzing,” Davies told United’s in-house media team after putting pen to paper.
“As soon as I heard about the opportunity it was something I was really looking forward to getting done.
“It took a bit of time to get done but I'm happy to be here. I spoke to my agent, the clubs went back and forth and got the deal sorted, so I'm buzzing to be here.”
Davies was at Bramall Lane to see the Blades beat Luton Town 2-0 on Saturday, after being granted permission to travel to Sheffield for a medical by Stoke.
“We're right in the mix to push for that play-off spot,” Davies, whose new club are eight points adrift of sixth-placed Huddersfield but with three games in hand, said.
“In this league, consistency is the key. The games come thick and fast so it's important we stick together, all work in the same direction and the results will come.
“Because the quality's here with the lads we've got.”