Past warm periods provide an opportunity to evaluate climate models under extreme forcing scenarios, in particular high (>800 ppmv) atmospheric CO2 concentrations. Although a post hoc intercomparison of Eocene (similar to 50 Ma) climate model simulations and geological data has been carried out previously, models of past high-CO2 periods have never been evaluated in a consistent framework. Here, we present an experimental design for climate model simulations of three warm periods within the early Eocene and the latest Paleocene (the EECO, PETM, and pre-PETM). Together with the CMIP6 pre-industrial control and abrupt 4 x CO2 simulations, and additional sensitivity studies, these form the first phase of DeepMIP - the Deep-time Model Intercomparison Project, itself a group within the wider Paleo-climate Modelling Intercomparison Project (PMIP). The experimental design specifies and provides guidance on boundary conditions associated with palaeogeography, greenhouse gases, astronomical configuration, solar constant, land surface processes, and aerosols. Initial conditions, simulation length, and output variables are also specified. Finally, we explain how the geological data sets, which will be used to evaluate the simulations, will be developed.