High-entropy-alloy (HEA) catalysts have been used in many challenging electrocatalytic reactions, e.g., CO2 reduction reaction (CO2RR) due to their promising properties. For CO2RR catalysts, tuning metal compositions in Cu-based catalysts is one of the techniques to control the desired products. Thus, this work investigated the optimal composition of Cu-Mn-Ni-Zn HEA catalysts using high-throughput screening (HTS) for CO2RR targeting on two competing routes toward CH4 and CH3OH products. The screening protocol evaluates catalytic activity through adsorption energy (Eads) of *CO2, *CO, *COOH, and *H. At the same time, the selectivity is represented by Eads of *COH, *CH4, *CHO, and *CH3OH, using density functional theory (DFT) accelerated by machine learning techniques. The screening result from 11,920 data revealed 259 candidates for CH4-selective and 4,214 for CH3OH-selective catalysts. Interestingly, the Cu-Mn-Ni-Zn excellently prevented competitive hydrogen evolution reaction by up to 90%. Optimal composition for each route are Cu0.1Mn0.4Ni0.2Zn0.3 and Cu0.2Mn0.4- Ni0.1Zn0.3 in CH4-selective route and Cu0.3Mn0.3Ni0.2Zn0.2, Cu0.3Mn0.2Ni0.3Zn0.2, Cu0.3Mn0.2Ni0.2Zn0.3, and Cu0.2Mn0.3Ni0.3Zn0.2 in CH3OH-selective route. The optimal catalyst structure with high CO2RR activity in both routes was revealed to have the Mn atom as an active site, while Cu, Ni, and Zn as neighboring atoms. Hence, the Cu-Mn-Ni-Zn HEA catalyst is the promising electrocatalyst for CO2RR.