Object tracking in RGB-thermal (RGB-T) videos is increasingly used in many fields due to the all-weather and all-day working capability of the dual-modality imaging system, as well as the rapid development of low-cost and miniaturized infrared camera technology. However, it is still very challenging to effectively fuse dual-modality information to build a robust RGB-T tracker. In this paper, an RGB-T object tracking algorithm based on a modal-aware attention network and competitive learning (MaCNet) is proposed, which includes a feature extraction network, modal-aware attention network, and classification network. The feature extraction network adopts the form of a two-stream network to extract features from each modality image. The modal-aware attention network integrates the original data, establishes an attention model that characterizes the importance of different feature layers, and then guides the feature fusion to enhance the information interaction between modalities. The classification network constructs a modality-egoistic loss function through three parallel binary classifiers acting on the RGB branch, the thermal infrared branch, and the fusion branch, respectively. Guided by the training strategy of competitive learning, the entire network is fine-tuned in the direction of the optimal fusion of the dual modalities. Extensive experiments on several publicly available RGB-T datasets show that our tracker has superior performance compared to other latest RGB-T and RGB tracking approaches.