The 23-year-old sustained the injury in the first half of last week's play-off semi-final second-leg defeat to Millwall and left Elland Road on crutches, wearing a protective boot.
Kilkenny said: "Over the weekend it was really swollen, so you couldn't tell the extent of the injury. I'm having an X-ray and I'll find out if it's a break, a fracture or a twist.
"It'll give me time so I won't miss any games and hopefully I'll be fit for pre-season."
The midfielder was on duty with the Australian Olympic squad in China last summer and struggled after missing pre-season.
He added: "When I came back last summer I wasn't performing to the level I was over the last 12 games or so, particularly when I was playing in central midfield, but hopefully I'll get fully fit and be ready to start again."