Fluid pressure changes affect fault stability and can promote the initiation of earthquakes and aseismic slip. However, the relationship between seismic and aseismic fault slip during fluid injection remains poorly understood. Here, we investigate, through 3‐D hydromechanical modeling, the spatiotemporal evolution of seismicity and aseismic slip on a permeable, slip‐weakening fault subjected to a local injection of fluid, under different prestress conditions. The model results in an expanding aseismic slip region, which concentrates shear stress at its edge and triggers seismicity. The aseismic slip dominates the slip budget, whatever the initial fault stress. We find that the seismicity is collocated with the aseismic rupture front rather than with the fluid pressure diffusion front. On faults initially far from failure, the aseismic rupture front is located behind or at the pressure front. On faults initially closer to failure, the model predicts that both the rupture front and the seismicity outpace the pressurized zone, resulting in a sharp increase of the migration velocity and released moment of the seismicity. Insights gained from this modeling study exhibit various features that are observed in sequences of induced earthquakes in both field experiments and natural reservoir systems and can help guide the interpretation of past and future observations of induced seismicity.