Speaker-attributed automatic speech recognition (ASR) in multi-speaker environments remains a significant challenge, particularly when systems conditioned on speaker embeddings fail to generalize to unseen speakers. In this work, we propose Diarization-Conditioned Whisper (DiCoW), a novel approach to target-speaker ASR that leverages speaker diarization outputs as conditioning information. DiCoW extends the pre-trained Whisper model by integrating diarization labels directly, eliminating reliance on speaker embeddings and reducing the need for extensive speaker-specific training data. Our method introduces frame-level diarization-dependent transformations (FDDT) and query-key biasing (QKb) techniques to refine the model's focus on target speakers while effectively handling overlapping speech. By leveraging diarization outputs as conditioning signals, DiCoW simplifies the workflow for multi-speaker ASR, improves generalization to unseen speakers and enables more reliable transcription in real-world multi-speaker recordings. Additionally, we explore the integration of a connectionist temporal classification (CTC) head to Whisper and demonstrate its ability to improve transcription efficiency through hybrid decoding. Notably, we show that our approach is not limited to Whisper; it also provides similar benefits when applied to the Branchformer model. We validate DiCoW on real-world datasets, including AMI and NOTSOFAR-1 from CHiME-8 challenge, as well as synthetic benchmarks such as Libri2Mix and LibriCSS, enabling direct comparisons with previous methods. Results demonstrate that DiCoW enhances the model's target-speaker ASR capabilities while maintaining Whisper's accuracy and robustness on single-speaker data.