Effective compression of 360-degree images, also referred to as omnidirectional images (ODIs), is of high interest for various virtual reality (VR) and related applications. 2D image compression methods ignore the equator-biased nature of ODIs and fail to address oversampling near the poles, leading to inefficient compression when applied to ODI. We present a new learned saliency-aware 360-degree image compression architecture that prioritizes bit allocation to more significant regions, considering the unique properties of ODIs. By assigning fewer bits to less important regions, significant data size reduction can be achieved while maintaining high visual quality in the significant regions. To the best of our knowledge, this is the first study that proposes an end-to-end variable-rate model to compress 360-degree images leveraging saliency information. The results show significant bit-rate savings over the state-of-the-art learned and traditional ODI compression methods at similar perceptual visual quality.