We performed direct numerical simulations (DNS) for the two-dimensional (2D) turbulent ignition of ultra-lean methane-air and n-heptane-air mixtures with a high exhaust gas recirculation (EGR) rate at high pressure to determine the ignition criteria and ignition delay time. We defined an initial high-temperature region as an ignition kernel and conducted one-dimensional preliminary DNS to determine the ignition criteria in terms of the ignition source energy and the thermal conduction from the ignition kernel during the induction period. Additionally, we analyzed the 2D DNS results to clarify the influence of the turbulent strain rate on the ignition delay time and the mechanism by which the turbulence influences the establishment of the ignition kernel. We observed that the distribution of eddies and the strain rate in the high-temperature region influences the success or failure of the ignition process and, therefore, the ignition delay time. The ignition delay time increases proportionally to the square of strain rate averaged in the high concentration region of the intermediate species during the induction period. This suggests that the ignition in a turbulent field is based on the balance between the influence of a locally averaged strain rate in the preheating region and the chemical (flame) time scale. Based on these observations, a simple model for the ignition delay time was constructed based on the mean strain rate in the high concentration region of the intermediate species during the induction period. The strain rate averaged in the high concentration region of the intermediate species was normalized by using the laminar burning velocity and the laminar thermal flame thickness. Additionally, the ignition delay time was normalized by the ignition delay time of the corresponding laminar case, yielding the same ignition model/criterion for both examined fuels, which could be extended to other mixtures.