The effect of the vector interaction on three-flavor magnetized matter is studied within the SU(3) Nambu-Jona-Lasinio quark model. We considered cold matter under a static external magnetic field within two different models for the vector interaction in order to investigate how the form of the vector interaction and the intensity of the magnetic field affect the equation of state as well as the strangeness content. It was shown that the flavor-independent vector interaction predicts a smaller strangeness content and, therefore, harder equations of state. On the other hand, the flavor-dependent vector interaction favors larger strangeness content the larger the vector coupling. We confirmed that, at low densities, the magnetic field and the vector interaction have opposite competing effects: the first one softens the equation of state while the second hardens it. Quark stars and hybrid stars subject to an external magnetic field were also studied. Larger star masses are obtained for the flavor-independent vector interaction. Hybrid stars may bare a core containing deconfined quarks if neither the vector interaction nor the magnetic field are too strong. Also, the presence of strong magnetic fields seems to disfavor the existence of a quark core in hybrid stars. © 2014 American Physical Society.