Deepfake speech utterances can be forged by replacing one or more words in a bona fide utterance with semantically different words synthesized with speech-generative models. While a dedicated synthetic word detector could be developed, we developed a cost-effective method that fine-tunes a pre-trained Whisper model to detect synthetic words while transcribing the input utterance via next-token prediction. We further investigate using partially vocoded utterances as the fine-tuning data, thus reducing the cost of data collection. Our experiments demonstrate that, on in-domain test data, the fine-tuned Whisper yields low synthetic-word detection error rates and transcription error rates. On out-of-domain test data with synthetic words produced with unseen speech-generative models, the fine-tuned Whisper remains on par with a dedicated ResNet-based detection model; however, the overall performance degradation calls for strategies to improve its generalization capability.