A scan has revealed a slight tear and United boss Kevin Blackwell says it could be at least two weeks before Wright is fit to play again.
The 28-year-old has already been ruled out of the friendlies with Valencia, today, Darlington and Hibs and is now setting his sights on the opening day with Derby on Saturday August 7.
Blackwell said in the Yorkshire Evening Post: "Wright could out for a couple of weeks. We're hoping he makes it for the first game of the season, but we'll see how things go. He felt his hamstring in the warm-up on Wednesday and I didn't think he was anywhere near as mobile as he can be.
"Maybe he shouldn't have played but such is the keeness among the players that they don't want to risk missing out and losing the shirt."
Matthew Spring suffered a groin strain in the same game and is a doubt for the Valencia visit.