The emergence of sixth-generation (6G) mobile networks and non-terrestrial networks (NTNs) has led to increased interest in low Earth orbit (LEO) satellite-based communication networks for their potential to provide global coverage and ubiquitous connectivity. In this paper, we investigate the LEO satellites selection-based computation offloading problem in the aircraft–satellite multi-access edge computing (ASMEC) network, where LEO satellites and edge computing processors are integrated to provide ubiquitous and low-latency communication and computation services for aircraft during flights. In contrast to most existing works, which directly assume a fixed number of satellites or orbits in satellite-based MEC networks, we investigate the problem of how many satellites and which satellites to select in the ASMEC network. Our objective is to minimize the average total time delay of tasks during aircraft–satellite computation offloading. To achieve this, we formulate a nonlinear integer programming (NLIP) problem and propose the LEO satellites selection-based computation offloading (LSSBCO) algorithm to solve it, which includes the shortest aircraft–satellite distance based access satellite selection (ASS-SD) algorithm and the nearest k(t) neighboring satellites selection (NSS-k(t)) algorithm. We evaluate the performance of the LSSBCO algorithm in terms of the average total time delay, the maximum throughput, the average aircraft–satellite distance, the average connection duration, and the number of satellite handovers. Numerical results show that the proposed algorithm outperforms the benchmark algorithms with a lower average total time delay.