Manager Gary Mills has a few selection issues with several of his squad not currently available.
The Minstermen are already without long-serving centre-back David McGurk and midfielder Scott Kerr, who are still recovering from knocks sustained from last season.
They have since been joined on the injury list by Lanre Oyebanjo, John McReady and Jamie Reed, although striker Matty Blair has recovered from a thigh strain.
“It’s one of those competitions that, with the luck of the draw, you can get a big boy by reaching the second round, which is great for finances but I want to go all the way," Mills told The Press.
"People say that’s not going to happen but funny things do happen in football and we have to go as far as we can. “People say going to Doncaster is a tough draw but we will go there confident because of the way we play and how positive we are and I think we can upset them."
Doncaster also have fitness worries and are checking on central defenders Liam Wakefield and Rob Jones, while latest signing Paul Quinn could earn a debut at right-back along with fellow newcomers David Cotterill, David Syers and Robbie Blake.
Rovers were relegated from the Championship at the end of last season after failing to win any of their final dozen home games.