As oil and gas exploration gradually advances into deep waters, the combined effects of various types of gas kick and the accurate calculation of the gas-kick volume have gained increasing attention. This study focused on gas kicks from permeable gas-bearing formations, considering the mass transfer of gas in the filtration region of the drilling fluids and revealed the mechanisms of seepage-driven and diffusion-driven gas kicks. Based on seepage mechanics and diffusion theory, a comprehensive model for calculating gas-kick volume was established, considering the synergistic effect of gas-concentration-diffusion and negative-differential-pressure, as well as mass transfer in both the filtrate zone and the filter-cake zone. The new model showed high calculation accuracy. The sensitivity analysis showed that both the seepage-driven and diffusion-driven gas-kick volumes in the wellbore increased with increasing formation porosity and open-hole length, while the thickness of the filter cake had a strong inhibitory effect on both. Additionally, a “seepage–diffusion ratio” was introduced to reveal the gas-kick evolution pattern under a seepage–diffusion mechanism. Under specific case conditions, when the seepage–diffusion ratio was less than approximately 1%, diffusion-driven gas kick contributed more than seepage-driven gas kick; when the seepage–diffusion ratio exceeded 1%, seepage-driven gas kick contributed more than diffusion-driven gas kick. The research can provide crucial parameters for wellbore multiphase flow calculation and wellbore pressure prediction.