Recent studies reveal a significant theoretical link between variational autoencoders (VAEs) and rate-distortion theory, notably in utilizing VAEs to estimate the theoretical upper bound of the information rate-distortion function of images. Such estimated theoretical bounds substantially exceed the performance of existing neural image codecs (NICs). To narrow this gap, we propose a theoretical bound-guided hierarchical VAE (BG-VAE) for NIC. The proposed BG-VAE leverages the theoretical bound to guide the NIC model towards enhanced performance. We implement the BG-VAE using Hierarchical VAEs and demonstrate its effectiveness through extensive experiments. Along with advanced neural network blocks, we provide a versatile, variable-rate NIC that outperforms existing methods when considering both rate-distortion performance and computational complexity. The code is available at BG-VAE.