The growing capabilities of Large Language Models (LLMs) have led to their widespread adoption for function completion within code repositories. Recent studies on such tasks show promising results when explicit instructions, often in the form of docstrings, are available to guide the completion. However, in real-world scenarios, clear docstrings are frequently absent. Under such conditions, LLMs typically fail to produce accurate completions. To enable more automated and accurate function completion in such settings, we aim to enable LLMs to accurately infer the developer's intent prior to code completion. Our key insight is that the preceding code, namely the code context before the function to be completed, often contains valuable cues that help the model understand the intended functionality. However, inferring intent from such implicit context is non-trivial and constitutes a core challenge in function-level code completion. To tackle this challenge, inspired by how humans interpret context, we propose a reasoning-based prompting framework that guides LLMs to utilize these contextual cues to infer intent step by step. To incentivize LLMs to reason through the preceding code and infer intent, we further curate a dataset of 40k examples, each annotated with intermediate reasoning traces and corresponding docstrings. Extensive experiments on DevEval and ComplexCodeEval demonstrate consistent performance improvements across multiple models, achieving over 25% relative gains in pass@1 for both DeepSeekCoder and CodeLLaMA families. Building upon our framework, we further develop an intent-interactive platform that supports lightweight human feedback. This platform allows developers to select from a set of candidate intentions or edit the intent to better guide the model. Our experiments show that this interactive approach leads to further performance improvements.