Quantum circuit mapping (QCM) is a crucial preprocessing step for executing a logical circuit (LC) on noisy intermediate-scale quantum (NISQ) devices. Balancing the introduction of extra gates and the efficiency of preprocessing poses a significant challenge for the mapping process. To address this challenge, we propose the mutual-influence-aware (MIA) heuristic method by integrating an initial mapping search framework, an initial mapping generator, and a heuristic circuit mapper. Initially, the framework utilizes the generator to obtain a favorable starting point for the initial mapping search. With this starting point, the search process can efficiently discover a promising initial mapping within a few bidirectional iterations. The circuit mapper considers mutual influences of SWAP gates and is invoked once per iteration. Ultimately, the best result from all iterations is considered the QCM outcome. The experimental results on extensive benchmark circuits demonstrate that, compared to the iterated local search (ILS) method, which represents the current state-of-the-art, our MIA method introduces a similar number of extra gates while achieving nearly 95 times faster execution.