Sky Italia says Udinese are monitoring developments at Deepdale.
The 25-year-old is out of contract at the end of the season and has long been a target for manager Chris Wilder as cover for Jack O'Connell.
They are already facing stiff competition from Scottish giants Celtic and Bournemouth, who it is believed submitted a £3m-£4m offer in the summer.
Chris Wilder and Prince Abdullah have both revealed that United's intention is to use the loan market, so unless there has been a change in approach, Davies could well be a target with a view to signing on a free in the summer.
However, should United which to pursue him on a permanent deal now, convincing him to come to Bramall Lane with Championship football and a squad player status a distinct possibility could be challenging, reports the Examiner.