Multi-channel inputs offer several advantages over singlechannel, to improve the robustness of on-device speech recognition systems. Recent work on multi-channel transformer, has proposed a way to incorporate such inputs into end-to-end ASR for improved accuracy. However, this approach is characterized by a high computational complexity, which prevents it from being deployed in on-device systems. In this paper, we present a novel speech recognition model, Multi-Channel Transformer Transducer (MCTT), which features end-to-end multi-channel training, low computation cost, and low latency so that it is suitable for streaming decoding in on-device speech recognition. In a far-field in-house dataset, our MCTT outperforms stagewise multi-channel models with transformer-transducer up to 6:01% relative WER improvement (WERR). In addition, MCTT outperforms the multi-channel transformer up to 11:62% WERR, and is 15:8 times faster in terms of inference speed. We further show that we can improve the computational cost of MCTT by constraining the future and previous context in attention computations.