Monte Carlo neutron transport codes have traditionally used a fixed-source scheme to simulate a subcritical system with an external source. The efficiency of this scheme is known to depend on the subcriticality level: The lower the subcriticality is, the worse is the efficiency. We have investigated an alternate iterative scheme, namely, the Monte Carlo iterative k-source (IKS) scheme, for the study of neutron subcritical multiplication. Our results show that the iterative scheme not only is as accurate, effective, and computationally efficient as the fixed-source scheme but also has the additional advantage of being weakly dependent on the subcriticality level. Also, the efficiency of this scheme is unaffected by the change in the location of the external source, unlike the fixed-source scheme where the efficiency decreases as the source is moved away from the fissile core center. The algorithm of this scheme is very similar to the algorithm of the eigenmode iterative scheme and hence can be easily implemented in the existing Monte Carlo codes. Our work establishes the validity and accuracy of the Monte Carlo IKS scheme, and with its incorporation in the production-level codes, it can be used for the physics design and analysis of accelerator-driven subcritical systems.