Advanced Siamese visual object tracking architectures are jointly trained using pair-wise input images to perform target classification and bounding box regression. They have achieved promising results in recent benchmarks and competitions. However, the existing methods suffer from two limitations: First, though the Siamese structure can estimate the target state in an instance frame, provided the target appearance does not deviate too much from the template, the detection of the target in an image cannot be guaranteed in the presence of severe appearance variations. Second, despite the classification and regression tasks sharing the same output from the backbone network, their specific modules and loss functions are invariably designed independently, without promoting any interaction. Yet, in a general tracking task, the centre classification and bounding box regression tasks are collaboratively working to estimate the final target location. To address the above issues, it is essential to perform target-agnostic detection so as to promote cross-task interactions in a Siamese-based tracking framework. In this work, we endow a novel network with a target-agnostic object detection module to complement the direct target inference, and to avoid or minimise the misalignment of the key cues of potential template-instance matches. To unify the multi-task learning formulation, we develop a cross-task interaction module to ensure consistent supervision of the classification and regression branches, improving the synergy of different branches. To eliminate potential inconsistencies that may arise within a multi-task architecture, we assign adaptive labels, rather than fixed hard labels, to supervise the network training more effectively. The experimental results obtained on several benchmarks, i.e., OTB100, UAV123, VOT2018, VOT2019, and LaSOT, demonstrate the effectiveness of the advanced target detection module, as well as the cross-task interaction, exhibiting superior tracking performance as compared with the state-of-the-art tracking methods.