The Blades have just lost shot-stopper Robin Olsen, who was on loan at the club, after he completed a switch to Premier League side Aston Villa.
Boss Paul Heckingbottom now wants to see the Blades swing into action to sign a replacement and they are turning their eyes towards Stoke.
Wales international Davies is the man on Sheffield United’s radar and the Blades want to sign him.
The shot-stopper has made 12 appearances for Stoke in the Championship so far this season, keeping five clean sheets along the way.
Davies came up against Sheffield United in the Championship in October as he conceded twice in a 2-1 loss for Stoke.
Heckingbottom has seen enough of the 29-year-old to believe he can fit in at Bramall Lane and he is a wanted man.
Davies came through the youth ranks at Sheffield United’s local rivals Sheffield Wednesday, but received his first team chances following a move to Barnsley.