The stable ferromagnetic ground state and controllable topological phase are important for the next-generation spintronic devices. Here we theoretically explored the possibility of the coexistence of long-range ferromagnetic coupling and topological order of narrow-gap PbS by late 3d transition-metal elements (Fe, Co, and Ni) doping and external strain. The results showed that such magnetically doping atom could produce high-spin polarized state. Ferromagnetic ground state was found for Fe- and Ni-doped PbS while antiferromagnetic ground state for the Co-doped case. Particularly, Ni doping could preserve the insulating nature of the host PbS, while the intrinsic band gap would remarkably decrease due to the strong p-d hybridization between Ni d and S p states. Further strain-driven band structure calculations suggested that an inverted band could be obtained at the lattice compressive strain of - 4% (pressure similar to 4.56 GPa), allowing one to switch from a regular to a topological insulator. Both the ferromagnetic and the topological insulating properties pointed out the possibility for realizing the potential quantum anomalous Hall effect in Ni-doped PbS.