Weakly supervised object localization (WSOL) aims to predict both object locations and categories with only image-level class labels. However, most existing methods rely on class-specific image regions for localization, resulting in incomplete object localization. To alleviate this problem, we propose a novel end-to-end task-aware framework with a transformer encoder-decoder architecture (TAFormer) to learn class-agnostic foreground maps, including a representation encoder, a localization decoder, and a classification decoder. The proposed TAFormer enjoys several merits. First, the designed three modules can effectively perform class-agnostic localization and classification in a task-aware manner, achieving remarkable performance for both tasks. Second, an optimal transport algorithm is proposed to provide pixel-level pseudo labels to online refine foreground maps. To the best of our knowledge, this is the first work by exploring a task-aware framework with a transformer architecture and an optimal transport algorithm to achieve accurate object localization for WSOL. Extensive experiments with four backbones on two standard benchmarks demonstrate that our TAFormer achieves favorable performance against state-of-the-art methods. Furthermore, we show that the proposed TAFormer provides higher robustness against adversarial attacks and noisy labels.