Highly swirling flows are often prone to precessing instabilities, with an azimuthal wave number of m=-1. We carry out a weakly nonlinear analysis to determine the response behavior of this instability to harmonic forcing. An incompressible flow is considered, where an annular inlet provides a swirling flow into a cylindrical region. For high swirl a vortex breakdown is induced, which is found to support an m=-1 instability. By expanding about the Reynolds number where this instability first occurs, a Stuart-Landau equation for the critical mode amplitude can be found and the effect of forcing can be assessed. Two types of forcing are considered. First, a Gaussian forcing confined to the inlet nozzle is used to study m=0 and m=-1 forcings. Second, optimal forcings (measured by the two-norm) with azimuthal wave numbers in the range -3≤m≤3 are considered. It is found that modal stabilization is highly dependent on the azimuthal wave number m, which governs whether the forcing is counter- or corotating with the direction of swirl. Counterrotating forcings are able to stabilize the mode for a wide range of forcing frequencies, while corotating forcings fail to yield a stable flow. In all cases, it is the base-flow modification induced by the forced response that is the dominant underlying feature responsible for the observed stabilization. This base-flow modification seeks to reduce axial momentum near the recirculation region for corotating forcings and increase it for counterrotating forcings, thus changing the size of the recirculation bubble and producing the two distinct response behaviors.