The England international has missed Villa's last two games with the injury, although Smith's men did beat Leeds without him at the weekend.
But the loss of a player that Blades boss Chris Wilder today described as the best player in England is a significant one for Villa.
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"He won't be ready for the game tomorrow, he's not travelling with us.
"We'll decide whether we can get him on the training ground later in the week. He's working with the doctor and the rehab staff every day.
"It's frustrating because he's a top player and we'd be a better team having him in it. But it's one of those things - players get injured. It's how you cope without those top players.
"He just feels it on one movement, so once that's free he can go and train. We're looking at whether we can get him on the training pitch by the end of this week."
Smith added: "There was a lot of noise outside that we were a one-man team and couldn't win without Jack. But we firmly believed in the squad of players we've got. But we also know we're a better team with Jack in it because of the top-class player he is.
"It's always disappointing when he's not in the team because of what he gives us, but we know we're more than capable of winning games without him."