The versatile defender was taken off on a stretcher during the 1-0 win against Tottenham on Monday after pulling up in pain as he chased a through ball by Larnell Cole.
Veseli, who was signed from Manchester City last year, has performed in a number of roles for Joyce's side and has adapted well to a holding midfield position.
“It’s his hamstring,” Joyce informed ManUtd.com. “It’s frustrating for him as he he’s had a good run of form recently and frustrating for us as well as we are quite thin on the ground.”
The news means that Veseli is out of the next league game - away to Wolves on 26 April - following the postponement of the Old Trafford fixture with Liverpool that was originally scheduled for Friday night.
Veseli was replaced by Tyler Blackett at the Salford City Stadium with the powerful defender slotting into a right-sided midfield role.