Most existing image-text matching methods adopt triplet loss as the optimization objective, and choosing a proper negative sample for the triplet of <anchor, positive, negative> is important for effectively training the model, e.g., hard negatives make the model learn efficiently and effectively. However, we observe that existing methods mainly employ the most similar samples as hard negatives, which may not be true negatives. In other words, the samples with high similarity but not paired with the anchor may reserve positive semantic associations, and we call them false negatives. Repelling these false negatives in triplet loss would mislead the semantic representation learning and result in inferior retrieval performance. In this paper, we propose a novel False Negative Elimination (FNE) strategy to select negatives via sampling, which could alleviate the problem introduced by false negatives. Specifically, we first construct the distributions of positive and negative samples separately via their similarities with the anchor, based on the features extracted from image and text encoders. Then we calculate the false negative probability of a given sample based on its similarity with the anchor and the above distributions via the Bayes' rule, which is employed as the sampling weight during negative sampling process. Since there may not exist any false negative in a small batch size, we design a memory module with momentum to retain a large negative buffer and implement our negative sampling strategy spanning over the buffer. In addition, to make the model focus on hard negatives, we reassign the sampling weights for the simple negatives with a cut-down strategy. The extensive experiments are conducted on Flickr30K and MS-COCO, and the results demonstrate the superiority of our proposed false negative elimination strategy. The code is available at https://github.com/LuminosityX/FNE.