Verifiable claim detection asks whether a claim expresses a factual statement that can, in principle, be assessed against external evidence. As an early filtering stage in automated fact-checking, it plays an important role in reducing the burden on downstream verification components. However, existing approaches to claim detection, whether based on check-worthiness or verifiability, rely solely on the claim text itself. This is a notable limitation for verifiable claim detection in particular, where determining whether a claim is checkable may benefit from knowing what entities and events it refers to and whether relevant information exists to support verification. Inspired by the established role of evidence retrieval in later-stage claim verification, we propose Context-Driven Claim Detection (ContextClaim), a paradigm that advances retrieval to the detection stage. ContextClaim extracts entity mentions from the input claim, retrieves relevant information from Wikipedia as a structured knowledge source, and employs large language models to produce concise contextual summaries for downstream classification. We evaluate ContextClaim on two datasets covering different topics and text genres, the CheckThat! 2022 COVID-19 Twitter dataset and the PoliClaim political debate dataset, across encoder-only and decoder-only models under fine-tuning, zero-shot, and few-shot settings. Results show that context augmentation can improve verifiable claim detection, although its effectiveness varies across domains, model architectures, and learning settings. Through component analysis, human evaluation, and error analysis, we further examine when and why the retrieved context contributes to more reliable verifiability judgments.