In this work, we utilize Large Language Models (LLMs) for a novel use case: constructing Performance Predictors (PP) that estimate the performance of specific deep neural network architectures on downstream tasks. We create PP prompts for LLMs, comprising (i) role descriptions, (ii) instructions for the LLM, (iii) hyperparameter definitions, and (iv) demonstrations presenting sample architectures with efficiency metrics and `training from scratch' performance. In machine translation (MT) tasks, GPT-4 with our PP prompts (LLM-PP) achieves a SoTA mean absolute error and a slight degradation in rank correlation coefficient compared to baseline predictors. Additionally, we demonstrate that predictions from LLM-PP can be distilled to a compact regression model (LLM-Distill-PP), which surprisingly retains much of the performance of LLM-PP. This presents a cost-effective alternative for resource-intensive performance estimation. Specifically, for Neural Architecture Search (NAS), we introduce a Hybrid-Search algorithm (HS-NAS) employing LLM-Distill-PP for the initial search stages and reverting to the baseline predictor later. HS-NAS performs similarly to SoTA NAS, reducing search hours by approximately 50%, and in some cases, improving latency, GFLOPs, and model size. The code can be found at: https://github.com/UBC-NLP/llmas.