Lennon stepped down from his position at Parkhead on May 22 following a successful four-year spell in Glasgow.
"I would like him to stay here for every reason under the sun. I think it would be great," O'Neill is quoted as saying by the Irish Herald.
"He is thinking things over, including a couple of other offers as well. I will do my very best to keep him. My very best.
"He wouldn't take [the decision] lightly. He loves it here, genuinely loves it here. It's been great for him. He has said it to me over the course of time and emphasised it. He loves it."
Keane played 13 games for Celtic before retiring from professional football in 2006.