Acoustic word embeddings (AWEs) are discriminative representations of speech segments, and learned embedding space reflects the phonetic similarity between words. With multi-view learning, where text labels are considered as supplementary input, AWEs are jointly trained with acoustically grounded word embeddings (AGWEs). In this paper, we expand the multi-view approach into a proxy-based framework for deep metric learning by equating AGWEs with proxies. A simple modification in computing the similarity matrix allows the general pair weighting to formulate the data-to-proxy relationship. Under the systematized framework, we propose an asymmetric-proxy loss that combines different parts of loss functions asymmetrically while keeping their merits. It follows the assumptions that the optimal function for anchor-positive pairs may differ from one for anchor-negative pairs, and a proxy may have a different impact when it substitutes for different positions in the triplet. We present comparative experiments with various proxy-based losses including our asymmetric-proxy loss, and evaluate AWEs and AGWEs for word discrimination tasks on WSJ corpus. The results demonstrate the effectiveness of the proposed method.