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Multilingual pre-trained language models (MPLMs) not only can handle tasks in different languages but also exhibit surprising zero-shot cross-lingual transferability. However, MPLMs usually are not able to achieve comparable supervised performance on rich-resource languages compared to the state-of-the-art monolingual pre-trained models. In this paper, we aim to improve the multilingual model's supervised and zero-shot performance simultaneously only with the resources from supervised languages. Our approach is based on transferring knowledge from high-performance monolingual models with a teacher-student framework. We let the multilingual model learn from multiple monolingual models simultaneously. To exploit the model's cross-lingual transferability, we propose MBLM (multi-branch multilingual language model), a model built on the MPLMs with multiple language branches. Each branch is a stack of transformers. MBLM is trained with the zero-shot-aware training strategy that encourages the model to learn from the mixture of zero-shot representations from all the branches. The results on two cross-lingual classification tasks show that, with only the task's supervised data used, our method improves both the supervised and zero-shot performance of MPLMs.