In this paper, we propose an invertible deep learning framework called INVVC for voice conversion. It is designed against the possible threats that inherently come along with voice conversion systems. Specifically, we develop an invertible framework that makes the source identity traceable. The framework is built on a series of invertible $1\times1$ convolutions and flows consisting of affine coupling layers. We apply the proposed framework to one-to-one voice conversion and many-to-one conversion using parallel training data. Experimental results show that this approach yields impressive performance on voice conversion and, moreover, the converted results can be reversed back to the source inputs utilizing the same parameters as in forwarding.