Twice-capped Wales international Davies joined the Blades from their Championship rivals Stoke City on Tuesday, signing an initial contract until the end of the season with the option to extend beyond then.
The 29-year-old made 15 appearances for the Potters in all competitions this season, keeping six clean sheets in the process, but was allowed to leave this month before becoming a free agent in the summer.
"As a professional athlete you want to play every week and be out on that pitch every Saturday and for the midweeks,” Davies said ahead of United’s trip to Peterborough on Saturday.
"I feel good, I have been working hard for a long time now. I’m looking forward to this opportunity to hopefully play some games.
"He’s been doing brilliant, Wes, Easty (Eastwood) as well. They’re really good goalkeepers.
"I’m here to provide healthy competition for all of us. It’s only going to push us better and make us better.
"I’ll be fully supportive of whoever’s playing and I’m sure they will be supportive of me if I’m in the sticks.”
Davies is already familiar to Blades boss Paul Heckingbottom, having worked under him at Barnsley from 2016 to 2018, where the pair won promotion from League One as well as the 2015/16 Football League Trophy.
Davies said: “I know all about him and he knows all about me.
"I know what he wants day in, day out. It’s going to be easy just to slot into the working day.”