Analogies play a central role in human commonsense reasoning. The ability to recognize analogies such as {``}eye is to seeing what ear is to hearing{’’}, sometimes referred to as analogical proportions, shape how we structure knowledge and understand language. Surprisingly, however, the task of identifying such analogies has not yet received much attention in the language model era. In this paper, we analyze the capabilities of transformer-based language models on this unsupervised task, using benchmarks obtained from educational settings, as well as more commonly used datasets. We find that off-the-shelf language models can identify analogies to a certain extent, but struggle with abstract and complex relations, and results are highly sensitive to model architecture and hyperparameters. Overall the best results were obtained with GPT-2 and RoBERTa, while configurations using BERT were not able to outperform word embedding models. Our results raise important questions for future work about how, and to what extent, pre-trained language models capture knowledge about abstract semantic relations.