Can my employer make me give a 30 day notice without any contract stating such?

I gave a verbal resignation notice to my supervisor on Tuesday (9/26/2016) and attempted to give him a letter; however, he stated that he would need to talk to his supervisor and get back to me. I told him I could work the majority of the next three weeks and set my resignation date as 10/14/2016. This morning, I, once again, attempted to give him the letter of resignation and was once again told he needed to talk to his supervisor. In this same conversation, he told me that the companies executive director stated that I have to give a 30 day notice. I have not signed anything to this effect. I have spoken with my HR department and was told that they would have to look into this. This is very frustrating as my new job starts on 10/17/2016. What are my options here? I do not want to be terminated and am afraid that if I refuse to give a 30 day notice that is what will happen.