Dual-atom catalysts (DACs) are promising materials for oxygen reduction reaction (ORR) and oxygen evolution reaction (OER) bifunctional electrocatalysts for rechargeable Zn-air batteries. Although there have been many theoretical studies of DACs, most of them assumed constant-charge conditions rather than the real electrochemical process of the constant-potential condition. In our work, we investigated Fe-based DACs for ORR and OER bifunctional catalytic performance by using constant-potential first-principles calculations. Among them, FeCo-N4-Gr exhibits high catalytic activity for the ORR and OER in both acidic and alkaline media. In addition, we demonstrated that the favorable electrochemical reaction pathways are accompanied by a regular change in the electronic properties of DACs. These fundamental insights provide guidelines for the design of ORR and OER bifunctional catalysts.