Davies, aged 29, is expected to be officially presented as a United player later this week, before Paul Heckingbottom’s side face Peterborough in the Championship.
Having previously worked with the United manager at Barnsley, the Wales international becomes Heckingbottom’s first signing of this month’s transfer window.
Although Wes Foderingham is poised to continue between the posts at London Road, Davies has been tasked with providing genuine competition for the former Rangers player as United look to qualify for the end-of-season play-offs.
Heckingbottom moved for Davies after learning that both the player and his parent club AS Roma wanted to end Robin Olsen’s loan and place him with Aston Villa instead.
The Swede, who completed his switch to the Midlands last week, was understood to be costing United around £30,000 a week in wages with the savings they have made used to finance the bid for his replacement.
Davies made over 200 appearances during his spell at Oakwell, before making another 41 senior outings after joining City. Fifteen of those have come this term, the last being against Preston North End on January 3rd.
Davies was granted permission to miss City’s meeting with Fulham on Saturday after confirming he was determined to push through the move to United and discovering Michael O’Neill, Heckingbottom’s counterpart at the bet365 Stadium was happy to grant his request.