Uncertainty characterization and propagation through computational models are the two key basic problems in risk and reliability analysis of structures and systems. Commonly used methods are mostly based on precise probability models, which are effective for characterizing the aleatory uncertainty. In real-world applications, the available data of model input variables commonly turn out to be scarce, incomplete and imprecise, and in this case, the epistemic uncertainty also emerges, which prevents us from generating the precise probability models. In this situation, the imprecise probability models such as probability-box model have been developed, and are shown to be especially useful for characterizing these two kinds of uncertainties in a unified framework. However, the performance of the available methods for propagating the imprecise probability models are generally computationally much more expensive than those developed for precise probability model, thus they are not widely used in practical applications. To fill this gap, a new general framework, termed as non-intrusive imprecise stochastic simulation, for efficiently propagating the imprecise probability models, and specifically, for estimating the failure probability bounds, has been developed. In this paper, we inject the line sampling method, which was originally developed for precise stochastic simulation, into the non-intrusive imprecise stochastic simulation framework, so as to further improve the efficiency when applied to low-nonlinear and high-dimensional problems, and to broaden the applicability of this framework. The computational cost of this new development is shown to be the same as the classical line sampling method. The effectiveness of the proposed framework is demonstrated by numerical test examples.