The midfielder is due to see out the current campaign at Bramall Lane after being deemed surplus to requirements at Derby County.
Having been allowed to head out on a short-term agreement, there would appear to be little chance of his expiring contract with the Rams being extended.
That will cast him into the free agent pool this summer and allow him to make a decision on his next move.
United are firm favourites to snap him up, with Davies having previously worked with Nigel Clough at Derby.
He is enjoying his time with the League One outfit and has offered an indication that he sees his future pushing for promotion in 2014/15.
Davies told the Sheffield Star in the wake of Tuesday's goalless draw with high-flying Brentford: "We know it's going to be difficult (this season) but we still believe we can make the play-offs.
"But there is still an eye on next season.
"We know these are the teams we've got to beat if we want to win the league and be right up there.
"It fills us with confidence in a way because the lads in the dressing room were saying that's the type of match we'd have lost one or two nil at the start of the season, and I thought we looked solid.
"It's not all about hard work. There's some fantastic talent here too.
"The squad as a whole is coming together nicely. Leyton Orient (on Saturday) will be another marker."