Text clustering aims to automatically partition a collection of text documents into distinct clusters based on linguistic features. In the literature, this task is usually framed as metric clustering based on text embeddings from pre-trained encoders or a graph clustering problem upon pairwise similarities from an oracle, e.g., a large ML model. Recently, large language models (LLMs) bring significant advancement in this field by offering contextualized text embeddings and highly accurate similarity scores, but meanwhile, present grand challenges to cope with substantial computational and/or financial overhead caused by numerous API-based queries or inference calls to the models. In response, this paper proposes TECL, a cost-effective framework that taps into the feedback from LLMs for accurate text clustering within a limited budget of queries to LLMs. Under the hood, TECL adopts our EdgeLLM or TriangleLLM to construct must-link/cannot-link constraints for text pairs, and further leverages such constraints as supervision signals input to our weighted constrained clustering approach to generate clusters. Particularly, EdgeLLM (resp. TriangleLLM) enables the identification of informative text pairs (resp. triplets) for querying LLMs via well-thought-out greedy algorithms and accurate extraction of pairwise constraints through carefully-crafted prompting techniques. Our experiments on multiple benchmark datasets exhibit that TECL consistently and considerably outperforms existing solutions in unsupervised text clustering under the same query cost for LLMs.