Despite the widespread use of statistical prior models in various fields, such models for neural network gradients have long been overlooked. The inherent challenge stems from their high-dimensional structures and complex interdependencies, which complicate effective modeling. In this work, we demonstrate the potential of large language models (LLMs) to act as gradient priors in a zero-shot setting. We examine the property by considering lossless gradient compression -- a critical application in distributed learning -- that depends heavily on precise probability modeling. To achieve this, we introduce LM-GC, a novel method that integrates LLMs with arithmetic coding.