The topological electronic properties of orthorhombic-phase Mo$_{2}$C and W$_{2}$C superconductors have been studied based on first-principles electronic structure calculations. Our studies show that both Mo$_{2}$C and W$_{2}$C are three-dimensional strong topological insulators defined on curved Fermi levels. The topological surface states on the (001) surface of Mo$_{2}$C right cross the Fermi level, while those of W$_{2}$C pass through the Fermi level with slight electron doping. These surface states hold helical spin textures and can be induced to become superconducting via a proximity effect, giving rise to an equivalent $p+ip$ type superconductivity. Our results show that Mo$_{2}$C and W$_{2}$C can provide a promising platform for exploring topological superconductivity and Majorana zero modes.
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