The pair are now due to hold transfer talks, although Jefferies has played down suggestions that terms have been finalised.
The 23-year-old stopper is looking for a new club following his release by Rangers.
"It's imperative that we have another goalkeeper signed for the new season, and I'll be talking to Lee about him joining us," he told the Kilmarnock Standard.
"Nothing has been agreed ahead of the meeting, but he's a young guy with a fair bit of experience under his belt."
Jefferies added on Rascle: "We had to accept the advice of the medical people after a second opinion came up with the same problem.
"It's a real shame, because Damien had done so well for us but it's just one of those unfortunate things that football throws up. We just couldn't take the risk."