Rather than commanding a fee, with a sum of £250,000 initially being discussed, sources last night reported that the goalkeeper convinced officials at bet365 Stadium to effectively terminate his contract - with United paying a nominal “undisclosed fee” to help facilitate the process.
That could explain why Davies is only initially tied to the club until the end of the campaign, although a longer term deal is set to be tabled in the summer.
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Aged 29, the Wales international underwent a medical at Bramall Lane earlier this week after being granted permission to miss City’s clash with Fulham by their manager Michael O’Neill.
Davies, who worked with Paul Heckingbottom at Barnsley before heading to Staffordshire, emerged as a target for United when the on-loan Robin Olsen and his parent club AS Roma submitted a request to terminate the supposedly season-long agreement they had entered into five months ago.
Although that deal could not be broken without United giving the green light, Heckingbottom allowed Olsen to complete a transfer to Aston Villa following talks with the 32-year-old.
After initially considering his City team mate Frank Fielding, United switched their attention to Davies when it became clear that O’Neill was prepared to let him leave.
The former Northern Ireland chief explained that Joe Bursik’s imminent return to action, coupled with Jack Bonham’s presence, meant he could afford to let Davies leave in search of regular first team football.
“It’s not really a decision that has been taken as a reflection on Adam,” O’Neill said, following the meeting with Marco Silva’s side.
After impressing in Olsen’s absence, Wes Foderingham has been told he will continue as United’s number one for the foreseeable future. However Davies and Jake Eastwood, who has been deputising for Foderingham since Olsen was ruled out by injury, are expected to provide him with genuine competition.
United also want to increase their options at centre-half before the window closes next week.
Williams was recently recalled to Anfield from his spell on loan with United’s Championship rivals Swansea City.