Although recent advances in deep learning technology have boosted automatic speech recognition (ASR) performance in the single-talker case, it remains difficult to recognize multi-talker speech in which many voices overlap. One conventional approach to tackle this problem is to use a cascade of a speech separation or target speech extraction front-end with an ASR back-end. However, the extra computation costs of the front-end module are a critical barrier to quick response, especially for streaming ASR. In this paper, we propose a target-speaker ASR (TS-ASR) system that implicitly integrates the target speech extraction functionality within a streaming end-to-end (E2E) ASR system, i.e. recurrent neural network-transducer (RNNT). Our system uses a similar idea as adopted for target speech extraction, but implements it directly at the level of the encoder of RNNT. This allows TS-ASR to be realized without placing extra computation costs on the front-end. Note that this study presents two major differences between prior studies on E2E TS-ASR; we investigate streaming models and base our study on Conformer models, whereas prior studies used RNN-based systems and considered only offline processing. We confirm in experiments that our TS-ASR achieves comparable recognition performance with conventional cascade systems in the offline setting, while reducing computation costs and realizing streaming TS-ASR.