For Convolutional Neural Networks (CNNs), Depthwise Separable CNN (DSCNN) is the preferred architecture for Application Specific Integrated Circuit (ASIC) implementation on edge devices. It benefits from a multi-mode approximate multiplier proposed in this work. The proposed approximate multiplier uses two 4-bit multiplication operations to implement a 12-bit multiplication operation by reusing the same multiplier array. With this approximate multiplier, sequential multiplication operations are pipelined in a modified DSCNN to fully utilize the Processing Element (PE) array in the convolutional layer. Two versions of Approximate-DSCNN (A-DSCNN) accelerators were implemented on TSMC 40 nm CMOS process with a supply voltage of 0.9 V. At a clock frequency of 200 MHz, the designs achieve 4.78 GOPs/mW and 4.89 GOP/mW power efficiency while occupying 1.16 mm2 and 0.398 mm2 area, respectively.