Former Celt Sylla is in the process of negotiating his release from Killie after several breaches of club discipline.
The 30-year-old Guinean was repeatedly pulled up by boss Jefferies for his time keeping and failure to pull his weight during pre-season and was eventually fined for his conduct.
But Jefferies finally had enough last week and at a meeting with the player, who only joined the club in January from Leicester City, it was agreed Sylla should move on.
Jefferies insists he can't afford to allow any of his squad preferential treatment as the key to Killie's recent success has been their unity.
He said: "There are certain standards we ask our players to adhere to and everyone is comfortable with them.
"You can't just go and do whatever you want. You have to ask permission.
"When you're running a football club you rely on the boys pulling together and we felt Momo did certain things that were unprofessional.
"I've told Momo he has no future here and there is no chance of that situation changing because we've already gone over things several times.
"He's apologised to me and has said himself it's maybe better if he leaves.
"I had to take a stand as I was in danger of losing my credibility with the rest of the players. They do what we ask them to do and they have no problems with that.
"It was basic things with Momo that were the problem - like his time keeping.
It's not possible to continually go over things when you've got someone who is being unprofessional.
"You can fine players and give them a warning but we'd done that with Momo and the problems persisted. In the end we reached the stage where it couldn't continue any longer.
"I was open with Momo and said if that's the way it's going to be then he's better finding a club who will put up with it because it's not going to happen at Kilmarnock.
"If he wants to keep doing his own thing then good luck to him. But he can do it somewhere else. At this club we rely on everyone pulling together as a unit - not as individuals.
"We had a talk and he agreed it was in the best interests of everyone if he left.
He's still under contract and we'll sort that out but we've not seen him since."
Jefferies is ready to hold further talks with Sylla's agent as the two parties attempt to come to an agreement over his release.
The manager added: "There are still a few things we need to tie up and I'm sure his agent will be in touch again soon. I had a chat with him last week and it wasn't for the first time.
"I still think Momo's a very good player. I don't know if he's had similar problems in the past.
"I had a chat with him, with Alex our physio as a witness, and said it couldn't continue and it was best he move on.
"We're not interested in individuals who want to do their own thing. You have to work within the squad here.
"He phoned me to apologise and said he may go back home. He apologised for being unprofessional and we left it at that.
"He can go where he likes. I don't have a problem with him going to another SPL club. But how he worked here wasn't what we were looking for.
"There is no one bigger than the club."