Radio Sheffield say the League Two side are looking to agree a season-long loan.
Marsh has been an unused substitute for each of the Blades' first three Premier League matches, and made his senior debut as a half-time substitute in the League Cup defeat at home to Lincoln City.
But with the Blades' attacking resources slowly being strengthened in the opening weeks, his parent club may now be in a position to loan out the 19-year-old attacking midfielder or forward.
Oli McBurnie has also returned from injury.
Doncaster's huge summer recruitment drive under new manager Grant McCann, which has brought Joe Ironside, Deji Sotona, Richard Wood, Jamie Sterry, Ian Lawlor, Rory Watson, Jack Senior, Owen Bailey, George Broadbent, Tom Nixon, Mo Faal and Tyler Roberts to the club is yet to yield results.
Rovers are bottom of League Two with one point from their opening five games and have only scored four goals.