The 27-year-old almost signed for the club in the January window, but Jim Jefferies was beaten by the transfer deadline.
With 26 goals to his name this season, Russell has also attracted interest from both Dundee United and clubs in England.
However, Jim Jefferies is confident a deal for local lad Russell will go through over the next few days.
"He had a few choices and he has let the other clubs know that his preferred option is Kilmarnock.
"It was frustrating that we ran out of time on the last day of the January transfer window but we made a commitment to him at the time that we would sign him when he was out of contract.
"In that period two or three clubs showed an interest. He had to do what was right, talk to them and think about it.
"The big advantage we had is he has got a good chance of going in the team and the other thing is he is from the area."
Recently named Second Division Player of the Year, Russell should g o some way to filling the void left by the sale of Colin Nish, who joined Hibs in January.