Ex-Owls boss Chris Turner, newly installed at County, wants to bring in the former Hartlepool defender Lee rating him as one of the best defenders in the division.
Lee has found times hard at Hillsborough under Sturrock because of injuries but played a big part in the team's clean sheet at Doncaster on Sunday.
Sturrock had earlier said that if Lee's contract had come to an end a month after his arrival at the club then he would have released the defender because of his fitness worries.
Branston meanwhile has had a mixed season for The Owls and at times looks dominating and then the next game looks shaky - like in the Hull City game for example.
Posh manager Barry Fry got Branston on loan from Rotherham last season and immediately installed him as captain because of his leadership qualities.
But despite the interest from both Turner and Fry, it is very unlikely Sturrock will let the duo leave at such an important time of the season - especially with Patrick Collins not being fully fit at the moment.
Meanwhile the Barnsley clash in early February will kick off at noon on police advice.