Recent advancements in diffusion models have enabled the generation of realistic deepfakes from textual prompts in natural language. While these models have numerous benefits across various sectors, they have also raised concerns about the potential misuse of fake images and cast new pressures on fake image detection. In this work, we pioneer a systematic study on deepfake detection generated by state-of-the-art diffusion models. Firstly, we conduct a comprehensive analysis of the performance of contrastive and classification-based visual features, respectively extracted from CLIP-based models and ResNet or ViT-based architectures trained on image classification datasets. Our results demonstrate that fake images share common low-level cues, which render them easily recognizable. Further, we devise a multimodal setting wherein fake images are synthesized by different textual captions, which are used as seeds for a generator. Under this setting, we quantify the performance of fake detection strategies and introduce a contrastive-based disentangling method that lets us analyze the role of the semantics of textual descriptions and low-level perceptual cues. Finally, we release a new dataset, called COCOFake, containing about 1.2M images generated from the original COCO image-caption pairs using two recent text-to-image diffusion models, namely Stable Diffusion v1.4 and v2.0.