The Blades have a gaping hole in the middle of the park that needs to be plugged after Sander Berge departed Bramall Lane.
They pocketed around £15 million from his move to Championship winners Burnley and could sign a replacement without spending a dime in transfer fees.
Davies is a free agent, having cut ties with his boyhood club Everton earlier this summer after his contract at Goodison Park ran out.
After 13 years on Merseyside, the English midfielder bid farewell to a club he joined as an 11-year-old and decided to explore other options.
The 25-year-old has been linked with a move to the Scottish Premiership, with Rangers previously keen to land his signature.
Manager Michael Beale is reportedly a fan, having expressed his admiration for the former England under-21 international earlier this year.
However, that transfer never materialised, and it now seems Davies could be on his way to Yorkshire, where he would link up with Paul Heckingbottom’s side.
That’s assuming United can convince the player to join the Bramall Lane project – although there shouldn’t be any obstacles to the deal, considering not many clubs are lining up to land his signature.
United have also been linked with a move for Southampton’s Will Smallbone, but the Saints are determined not to sell.
The Championship club have already lost James Ward-Prowse this summer, while Romeo Lavia is on the cusp of following his former skipper through the exit door.
Smallbone has proven his value, and the Saints want him to continue his development on the south coast despite having just one year left on his contract.
The player is currently injured, which could be a blessing in disguise for Southampton as interested parties are unlikely to pursue a move for a player in the treatment room.
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