The local field enhancement in proximity of metallic nanostructures can strongly modify the excitation and emission behaviors for the nearby fluorophore. In this paper, Maxwell's time-dependent curl equations are solved by the finite-difference time-domain method to investigate the electric field enhancement around an atomic-force microscopy (AFM) tip and a Au nanosphere (NS). To lower the background fluorescence signal, we proposed to induce the fluorescence quenching by placing the emitter at an optimized position that is 2 nm away from the Au NS. The AFM tip is thereby moved to the vicinity of the emitter quenched by the Au NS. The fluorescence enhancement factor (FEF) increases rapidly when the tip approaches the Au NS. A maximum FEF of 1500-fold is obtained when their separation is 4 nm. By laterally scanning the tip over the Au NS at a constant height, the full width at half-maximum of fluorescence's signal peak with respect to tip position is around 20 nm. This high sensitivity of the FEF on the relative position of the tip and Au NS provides valuable information to guide future experiments on high-resolution optical imaging and fluorescence enhancement for high quantum yield emitters.