Hay's contract expires at the end of June but the 35-year-old had been promised a role on Kenny Shiels' coaching staff.
However, the Northern Irishman has since left the club and Hay has also been told he has no future at Rugby Park.
"I had met Michael Johnston when the season finished but nothing had been decided," Hay said.
"Then I spoke to him again and he told me that there was nothing there for me.
"I automatically checked the list to see who Killie got on the opening day, saw it was Aberdeen away - and then realised that I wouldn't be involved.
"That's bad enough but I really don't know how I'm going to be able to tell my son that his daddy won't be playing for Kilmarnock again.
"Blake is five and he supports the club because it's in his blood, the same way it was in mine."
Hay signed for Kilmarnock in 1995 and made his debut against Rangers on the opening day of the 1999-2000 season and had his testimonial three years ago.