The 26-year-old former Dunfermline Athletic player, who joined Killie in 2003, has started this season with four goals in his opening 11 appearances.Killie manager Jim Jefferies confirmed: "He has knocked back our offer. That's the world we live in now and players can do that.
"It might be the case that he will change his mind or decide to stay if no offers come in."
Nish netted 14 times last season for Kilmarnock.
Kilmarnock have lost two of their prolific scorers, Kris Boyd and Steven Naismith, to Rangers in the last two seasons.
The Ayrshire club were keen to secure Nish beyond the end of his current contract, which expires at the end of the season.