The former Republic of Ireland Under-21 international landed awkwardly against Livingston last weekend and could face two months out, although manager Jim Goodwin was still waiting on the exact prognosis when he updated the media.
"Unfortunately the results of the scan weren't great so he is due to see a specialist later, and there is a possibility we will need to pencil him in for an operation next week," Goodwin said.
"He has done two or three different things to the ankle. It's not something we think is going to keep him out for three or four months. I would imagine it will be the six-to-eight-week bracket.
"He is a young lad, he has never had any serious injuries. He is physically strong and I would expect him to heal quickly.
"The thought of going into surgery now is not as big a deal as it used to be. We are hoping for as good a turnaround as possible."
Goodwin had a ready-made replacement on the bench against Livi with former Motherwell defender Charles Dunne slotting in seamlessly to the back three.
Ahead of Saturday's cinch Premiership clash with Hearts at Tynecastle, Goodwin added: "Conor has been a massive player for us, he has been very consistent and of course he will be a big loss.
"But at the same time Charles Dunne came on last week and was excellent. We have a very, very good replacement.
"Physically I think Charles is one of the outstanding centre-backs in the league, in terms of his all-round profile - 6ft 3in, rapid, technically very, very good.
"It's been well documented that Dunney's issues have been staying fit. That's why he hasn't been able to get back in the team. He started the season very well, picked up a knock and has found himself on the sidelines because the three lads who have been playing have been very consistent. So Charles has had to be patient.
"He came on Saturday and was excellent. We are very lucky to have him. It's not like I am turning to an 18 or 19-year-old replacement. He has a wealth of experience and all the ability in the world.”