Nunez is currently a free transfer having seen his contract cancelled by Uruguayan outfit Estudiantes La Plata (whom also have Juan Veron on their books) are is keen to sign for an English Premiership side.
The Paraguayan international signed from Racing Club (Argentina) in 2005 and has made 19 caps for his Country, helping them reach the last sixteen in this years World Cup in Germany.
The Blades have apparently jumped the queue for Nunez after offering the Defender a trial at Bramall Lane.
Mainly a Left-Back the 28 year old would be able to sign for a club immediately without waiting until the January transfer window.
Meanwhile Sheffield United boss Neil Warnock is one of a number of clubs who play to take striker David Healy away from Leeds United in January.
The Elland Road club are currently cash strapped and could be forced to sell their prize asset (including Matthew Kilgallon) with three un-named clubs interested in Healy following his hat-trick against Spain for North Ireland.