The Scottish international has been given permission to speak with Blades boss, Bryan Robson after Jagielka's £4m move to Everton was confirmed following weeks of waiting for the independent arbitration panel's decision to uphold that of the FA Premier League to only fine West Ham United and not dock the Hammers points for the Carlos Tevez affair.
Naysmith's possible switch to Bramall Lane has been mooted in press speculation but appeared unlikely when it emerged that the two clubs were a fair distance apart on the defender's value.
Naysmith found it difficult to hold down a first-team place at Goodison in the face of competition from Nuno Valente, Phil Neville and, of course, Joleon Lescott in that position last season, the latter excelling last term in what was for him an unnatural role.