We propose a dual-path self-attention recurrent neural network (DP-SARNN) for time-domain speech enhancement. We improve dual-path RNN (DP-RNN) by augmenting inter-chunk and intra-chunk RNN with a recently proposed efficient attention mechanism. The combination of inter-chunk and intra-chunk attention improves the attention mechanism for long sequences of speech frames. DP-SARNN outperforms a baseline DP-RNN by using a frame shift four times larger than in DP-RNN, which leads to a substantially reduced computation time per utterance. As a result, we develop a real-time DP-SARNN by using long short-term memory (LSTM) RNN and causal attention in inter-chunk SARNN. DP-SARNN significantly outperforms existing approaches to speech enhancement, and on average takes 7.9 ms CPU time to process a signal chunk of 32 ms.