Acoustic word embeddings (AWEs) are fixed-dimensional representations of variable-length speech segments. For zero-resource languages where labelled data is not available, one AWE approach is to use unsupervised autoencoder-based recurrent models. Another recent approach is to use multilingual transfer: a supervised AWE model is trained on several well-resourced languages and then applied to an unseen zero-resource language. We consider how a recent contrastive learning loss can be used in both the purely unsupervised and multilingual transfer settings. Firstly, we show that terms from an unsupervised term discovery system can be used for contrastive self-supervision, resulting in improvements over previous unsupervised monolingual AWE models. Secondly, we consider how multilingual AWE models can be adapted to a specific zero-resource language using discovered terms. We find that self-supervised contrastive adaptation outperforms adapted multilingual correspondence autoencoder and Siamese AWE models, giving the best overall results in a word discrimination task on six zero-resource languages.