Ship-radiated noise is the main basis for ship detection in underwater acoustic environments. Due to the increasing human activity in the ocean, the captured ship noise is usually mixed with or covered by other signals or noise. On the other hand, due to the softening effect of bubbles in the water generated by ships, ship noise undergoes non-negligible nonlinear distortion. To mitigate the nonlinear distortion and separate the target ship noise, blind source separation (BSS) becomes a promising solution. However, underwater acoustic nonlinear models are seldom used in research for nonlinear BSS. This paper is based on the hypothesis that the recovery and separation accuracy can be improved by considering this nonlinear effect in the underwater environment. The purpose of this research is to explore and discover a method with the above advantages. In this paper, a model is used in underwater BSS to describe the nonlinear impact of the softening effect of bubbles on ship noise. To separate the target ship-radiated noise from the nonlinear mixtures, an end-to-end network combining an attention mechanism and bidirectional long short-term memory (Bi-LSTM) recurrent neural network is proposed. Ship noise from the database ShipsEar and line spectrum signals are used in the simulation. The simulation results show that, compared with several recent neural networks used for linear and nonlinear BSS, the proposed scheme has an advantage in terms of the mean square error, correlation coefficient and signal-to-distortion ratio.