if I designed my own ASIC machine, what is the easiest way to connect it with the mining pool? I was searching but couldn't find any pool that has an API to communicate.
Mining pools use protocols such as Stratum to communicate.
However, I'd find it unusual to implement this in firmware or in an ASIC. I'd expect the ASICs to be dedicated to hashing and have pool communication handled by a supervisory computer. But I've not designed ASICs or mining systems so maybe my intuition here is weak.