McBurnie, who has scored six goals in the league this season, will miss United's remaining four matches, manager Chris Wilder told reporters on Friday.
"It's a massive blow for us, but it goes alongside a lot of the situations we've had all season," Wilder said.
"It doesn't require surgery, but obviously that's a big one to miss from a leadership and experience point of view."
James McAtee, on loan from Manchester City, is also a doubt with a calf injury after making 29 league appearances for United this season.
"McAtee went for a scan and we’re waiting for the results," Wilder added.
Sheffield United, who have 16 points from 34 matches, are 10 points below the safety zone and will be relegated if they lose at Newcastle United on Saturday.
"Through good times, we all enjoy it and we are all willing to take that credit when it happens," Wilder said.
"And you have to own and suffer and take responsibility for the times when there is a little bit of pain.
"As a club, there is humility. A lot of humble people. We've had a poor season, that's there for all to see."
