Traditional machine learning (ML) models usually rely on large-scale labeled datasets to achieve strong performance. However, such labeled datasets are often challenging and expensive to obtain. Also, the predefined categories limit the model's ability to generalize to other visual concepts as additional labeled data is required. On the contrary, the newly emerged multimodal model, which contains both visual and linguistic modalities, learns the concept of images from the raw text. It is a promising way to solve the above problems as it can use easy-to-collect image-text pairs to construct the training dataset and the raw texts contain almost unlimited categories according to their semantics. However, learning from a large-scale unlabeled dataset also exposes the model to the risk of potential poisoning attacks, whereby the adversary aims to perturb the model's training dataset to trigger malicious behaviors in it. Previous work mainly focuses on the visual modality. In this paper, we instead focus on answering two questions: (1) Is the linguistic modality also vulnerable to poisoning attacks? and (2) Which modality is most vulnerable? To answer the two questions, we conduct three types of poisoning attacks against CLIP, the most representative multimodal contrastive learning framework. Extensive evaluations on different datasets and model architectures show that all three attacks can perform well on the linguistic modality with only a relatively low poisoning rate and limited epochs. Also, we observe that the poisoning effect differs between different modalities, i.e., with lower MinRank in the visual modality and with higher Hit@K when K is small in the linguistic modality. To mitigate the attacks, we propose both pre-training and post-training defenses. We empirically show that both defenses can significantly reduce the attack performance while preserving the model's utility.