The rapid growth of data traffic and the emerging AI-native wireless architectures in NextG cellular systems place new demands on the fronthaul links of Cloud Radio Access Networks (C-RAN). In this paper, we investigate neural compression techniques for the Common Public Radio Interface (CPRI), aiming to reduce the fronthaul bandwidth while preserving signal quality. We introduce two deep learning-based compression algorithms designed to optimize the transformation of wireless signals into bit sequences for CPRI transmission. The first algorithm utilizes a non-linear transformation coupled with scalar/vector quantization based on a learned codebook. The second algorithm generates a latent vector transformed into a variable-length output bit sequence via arithmetic encoding, guided by the predicted probability distribution of each latent element. Novel techniques such as a shared weight model for storage-limited devices and a successive refinement model for managing multiple CPRI links with varying Quality of Service (QoS) are proposed. Extensive simulation results demonstrate notable Error Vector Magnitude (EVM) gains with improved rate-distortion performance for both algorithms compared to traditional methods. The proposed solutions are robust to variations in channel conditions, modulation formats, and noise levels, highlighting their potential for enabling efficient and scalable fronthaul in NextG AI-native networks as well as aligning with the current 3GPP research directions.