Phrase grounding, the problem of associating image regions to caption words, is a crucial component of vision-language tasks. We show that phrase grounding can be learned by optimizing word-region attention to maximize a lower bound on mutual information between images and caption words. Given pairs of images and captions, we maximize compatibility of the attention-weighted regions and the words in the corresponding caption, compared to non-corresponding pairs of images and captions. A key idea is to construct effective negative captions for learning through language model guided word substitutions. Training with our negatives yields a ~10% absolute gain in accuracy over randomly-sampled negatives from the training data. Our weakly supervised phrase grounding model trained on COCO-Captions shows a healthy gain of 5.7% to achieve 76.7% accuracy on Flickr30K Entities benchmark.