Voice conversion with deep neural networks has become extremely popular over the last few years with improvements over the past VC architectures. In particular, GAN architectures such as the cycleGAN and the VAEGAN have offered the possibility to learn voice conversion from non-parallel databases. However, GAN-based methods are highly unstable, requiring often a careful tuning of hyper-parameters, and can lead to poor voice identity conversion and substantially degraded converted speech signal. This paper discusses and tackles the stability issues of the GAN in the context of voice conversion. The proposed SoftGAN method aims at reducing the impact of the generator on the discriminator and vice versa during training, so both can learn more gradually and efficiently during training, in particular avoiding a training not in tandem. A subjective experiment conducted on a voice conversion task on the voice conversion challenge 2018 dataset shows that the proposed SoftGAN significantly improves the quality of the voice conversion while preserving the naturalness of the converted speech.