We investigate the potential of the International Linear Collider (ILC) to probe the mechanisms of neutrino mass generation and leptogenesis within the minimal seesaw model. Our results can also be used as an estimate for the potential of a Compact Linear Collider (CLIC). We find that heavy sterile neutrinos that simultaneously explain both, the observed light neutrino oscillations and the baryon asymmetry of the universe, can be found in displaced vertex searches at ILC. We further study the precision at which the flavour-dependent active-sterile mixing angles can be measured. The measurement of the ratios of these mixing angles, and potentially also of the heavy neutrino mass splitting, can test whether minimal type I seesaw models are the origin of the light neutrino masses, and it can be a first step towards probing leptogenesis as the mechanism of baryogenesis. Our results show that the ILC can be used as a discovery machine for New Physics in feebly coupled sectors that can address fundamental questions in particle physics and cosmology.