Clarke picked up the injury during Sunday's 2-0 home win over Chesterfield.
The Bantams should find out the exact length of time Clarke will be out for after the injury is operated on but he is certainly expected to miss both legs of City's forthcoming Sky Bet League 1 Play-Off semi-final with Millwall.
Speaking about Clarke's injury, Bantams boss Phil Parkinson said: "Billy got a whack on his cheekbone on Sunday and he went straight to the hospital after the game to get it checked out.
"Unfortunately for him, there's a depressed fracture of his cheekbone and he's having it operated on today (Tuesday).
"We should know more about how long he will be sidelined for after the operation."
Despite the loss of Clarke, Parkinson will at least be boosted by the return of fellow forward Steven Davies for the Play-Offs after the 28 year-old completed a three-match ban at the weekend.
James Hanson will have a scan this week on the calf injury that ruled him out of the Chesterfield win.
City will also monitor Reece Burke in the build up to Sunday's first leg against Millwall after the young defender came off at the weekend with a groin strain.