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 critical for a quick response, especially for streaming ASR. In this paper, we propose a target-speaker ASR (TS-ASR) system, which integrates implicitly 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 target speech extraction, but implements it directly at the level of the encoder of RNNT. This allows to realize TS-ASR without extra computation costs for the front-end. Note that our works present two major differences between prior studies about E2E-TS-ASR, we investigate streaming models and base our study on Conformer models, whereas prior studies used RNN-based systems and only dealt with offline processing. We confirm in experiments that our TS-ASR achieves comparable recognition performance with conventional cascade system in offline setting, while reducing computation costs and allowing streaming TS-ASR.