Multi-speaker automatic speech recognition (ASR) aims to transcribe conversational speech involving multiple speakers, requiring the model to capture not only what was said, but also who said it and sometimes when it was spoken. Recent Speech-LLM approaches have shown the potential of unified modeling for this task, but jointly learning speaker attribution, temporal structure, and lexical recognition remains difficult and data-intensive. At the current stage, leveraging reliable speaker diarization as an explicit structural prior provides a practical and efficient way to simplify this task. To effectively exploit such priors, we propose DM-ASR, a diarization-aware multi-speaker ASR framework that reformulates the task as a multi-turn dialogue generation process. Given an audio chunk and diarization results, DM-ASR decomposes transcription into a sequence of speaker- and time-conditioned queries, each corresponding to one speaker in one time segment. This formulation converts multi-speaker recognition into a series of structured sub-tasks, explicitly decoupling speaker-temporal structure from linguistic content and enabling effective integration of diarization cues with the reasoning capability of large language models. We further introduce an optional word-level timestamp prediction mechanism that interleaves word and timestamp tokens, yielding richer structured outputs and better transcription quality. Our analysis shows that diarization systems provide more reliable speaker identities and segment-level boundaries, while LLMs excel at modeling linguistic content and long-range dependencies, demonstrating their complementary strengths. Experiments on Mandarin and English benchmarks show that the proposed approach achieves strong performance with relatively small models and training data, while remaining competitive with or outperforming existing unified approaches.