Conversational Speech Synthesis (CSS) aims to effectively take the multimodal dialogue history (MDH) to generate speech with appropriate conversational prosody for target utterance. The key challenge of CSS is to model the interaction between the MDH and the target utterance. Note that text and speech modalities in MDH have their own unique influences, and they complement each other to produce a comprehensive impact on the target utterance. Previous works did not explicitly model such intra-modal and inter-modal interactions. To address this issue, we propose a new intra-modal and inter-modal context interaction scheme-based CSS system, termed III-CSS. Specifically, in the training phase, we combine the MDH with the text and speech modalities in the target utterance to obtain four modal combinations, including Historical Text-Next Text, Historical Speech-Next Speech, Historical Text-Next Speech, and Historical Speech-Next Text. Then, we design two contrastive learning-based intra-modal and two inter-modal interaction modules to deeply learn the intra-modal and inter-modal context interaction. In the inference phase, we take MDH and adopt trained interaction modules to fully infer the speech prosody of the target utterance's text content. Subjective and objective experiments on the DailyTalk dataset show that III-CSS outperforms the advanced baselines in terms of prosody expressiveness. Code and speech samples are available at https://github.com/AI-S2-Lab/I3CSS.