Non-autoregressive automatic speech recognition (ASR) has become a mainstream of ASR modeling because of its fast decoding speed and satisfactory result. To further boost the performance, relaxing the conditional independence assumption and cascading large-scaled pre-trained models are two active research directions. In addition to these strategies, we propose a lexical-aware non-autoregressive Transformer-based (LA-NAT) ASR framework, which consists of an acoustic encoder, a speech-text shared encoder, and a speech-text shared decoder. The acoustic encoder is used to process the input speech features as usual, and the speech-text shared encoder and decoder are designed to train speech and text data simultaneously. By doing so, LA-NAT aims to make the ASR model aware of lexical information, so the resulting model is expected to achieve better results by leveraging the learned linguistic knowledge. A series of experiments are conducted on the AISHELL-1, CSJ, and TEDLIUM 2 datasets. According to the experiments, the proposed LA-NAT can provide superior results than other recently proposed non-autoregressive ASR models. In addition, LA-NAT is a relatively compact model than most non-autoregressive ASR models, and it is about 58 times faster than the classic autoregressive model.