Maritime transport was responsible for 2.9% of anthropogenic global CO2 emissions in 2018, emitting 1,076 million tons of CO2. In the EU, marine transportation accounted for 3-4% of total CO2 emissions and emitted over 144 million tons of CO2 in 2019. In order to reduce greenhouse gas emissions from maritime transport, the European Union has mandated a reduction by 2% as of 2025 and an 80% reduction by 2050 (compared to 2020). One of the ways to achieve the targets is to use renewable fuels such as hydrogen and methanol in marine engines. However, due to their high auto-ignition temperature, they cannot be used as a single fuel in the compression-ignition engines, and diesel pilot injection is required to start ignition. Therefore, the co-combustion of hydrogen-diesel and methanol-diesel in the resulting dual-fuel engines should be well understood. To accurately simulate the combustion of hydrogen-diesel and methanol-diesel in dual-fuel mode, the ignition delay (ID) of the pilot fuel should be carefully estimated in the presence of premixed fuel. In the present work, the ignition delay (ID) for pilot ignited dual-fuel engine operation with hydrogen or methanol is investigated. A constant volume batch reactor numerical setup is used in the open source Cantera code to calculate the effects of premixed fuel on the ID of the pilot fuel. First, suitable chemical kinetic schemes are selected based on experimental data from the literature. Then ID is estimated for different engine-like operating conditions. The effects of different proportions of hydrogen and methanol, as well as different lambdas, on ID in the high-pressure regime are considered. It is shown how ID of the pilot fuel is strongly affected by the presence of premixed fuel.