Is it possible a post-doc position for a specialist in medicine without PhD? I graduated from Medical Faculty ten years ago. I got my degree as a Microbiology Specialist in 2015. I have been working as a Clinical Microbiologist in a hospital since then. I completed all my education and working in Turkey.

Last month, I attended a conference. I met a professor from King's College London. After discussing some issues about bacterial interactions this professor offered me a postdoc position in his department. I want to accept this position but I don't have a Ph.D. degree. The professor also doesn't sure and he advised me to search this issue. 

Does anyone know if it is possible or can you address me an organization to ask this question?
The organization to ask is the organization that wants to employ you. The name postdoc suggest that a postdoc starts after you became a doctor, but there is often some leeway in interpreting those rules. However, how much leeway depends a lot on the specific country and institution. Even within institutions departments can differ quite a lot in the extend to which they are willing to make use of the leeway offered by the rules set at the higher levels. So the only advise we can give you is to talk to HR department of the university that wants to hire you.