The 24-year-old right-back is likely to leave Everton after a bit-part role at the club.
He has been on the Blades' radar for some time and was considered in January before the club signed Ki-Jana Hoever on loan from Wolves.
However, the Dutchman failed to challenge Femi Seriki for his place, making a right-back a summer priority for manager Chris Wilder.
A return for Sheffield United hero Kyle Walker is also being considered, but the former Manchester City star would need to reduce his wage demands significantly.
The Blades aren't the only club interested in Patterson, with Serie A side Bologna also reportedly keen on the Scotland international.
Former Scotland striker Paul Dickov recently spoke highly of Patterson, saying: "Nathan Patterson is at the age where you have to be playing football.
"He's had some unfortunate injuries but now that he's back, he needs game time.
"There's no doubting his ability; he's a top young defender."
He added: "If he's available, I'm sure there'll be plenty of clubs interested in him.
"But I think a move back to Rangers, where he's comfortable and in surroundings he knows, could reignite his career."
However, Rangers' interest could complicate the situation if they decide to make a move for their former player, while Spanish club Sevilla have also been linked previously.