Let me see if I get your question. Are you saying that you have your CPU cooler attached to your current i7-6700K on an OLD motherboard and you want to transfer BOTH the CPU Cooler and the i7-6700K on a NEW motherboard?
If so, then you cannot remove the i7-6700K from your old motherboard without detaching its CPU Cooler. It would be recommended to clean out the old thermal paste and re-apply new once you have transferred both your i7-6700K and your CPU Cooler to the new motherboard.