Although the deep integration of the Automatic Speech Recognition (ASR) system with Large Language Models (LLMs) has significantly improved accuracy, the deployment of such systems in low-latency streaming scenarios remains challenging. In this paper, we propose Uni-ASR, a unified framework based on LLMs that integrates both non-streaming and streaming speech recognition capabilities. We propose a joint training paradigm that enables the system to seamlessly transition between two recognition modes without any architectural modifications. Furthermore, we introduce a context-aware training paradigm and a co-designed fallback decoding strategy, which can enhance streaming recognition accuracy without introducing additional latency. The experimental results demonstrate that Uni-ASR not only achieves competitive performance within non-streaming mode, but also demonstrates strong effectiveness in streaming scenarios under diverse latency constraints.