The 35-year-old had been promised a new contract as player-coach by Kenny Shiels before the Northern Irishman lost his job as manager.
Hay said in a number of national newspapers: "I had met Michael Johnston once the season had finished but nothing had been decided.
"Then I spoke to him on the phone and he told me that there was nothing there for me."
Hay joined his hometown team as a youth player in 1995, making his first-team debut in 1999, and he admits he is finding it hard to digest the news.
Hay said: "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."