The Sun says the Blades will ask United for Rojo on loan.
Blades boss Chris Wilder is looking for a new defender. The Argentinian centre-half is on the fringes at Old Trafford.
And he could be available in January.
Rojo, 30, was the subject of inquiries from several clubs during the last transfer window.
But United kept him and put him in their squad for the Champions League.
The fiery defender returned from a disappointing loan back home with Estudiantes in Argentina last term.