The physical origin of fast radio bursts (FRBs) is still unclear. However, young magnetars associated with short-duration gamma-ray bursts (SGRBs) have been thought to be possible central engines for some FRBs. In this paper, we perform a systematic search for SGRBs that are associated with FRBs in a sample including 623 FRBs (601 one-off bursts and 22 repeaters) and 168 SGRBs with precise localizations. We find that FRB 190309A is spatially associated with GRB 060502B, with a chance probability of 0.05 when temporal and redshift information is taken into account. Considering the high chance probability (the statistical significance is <3{sigma}), we examine other observational properties such as the host galaxy, the dispersion measure, and the energy budget of the central engine to check the possibility of their association. Although the available observational information is insufficient to determine whether they are physically associated, it does not rule out such a possibility. As the only pair of FRB and GRB that are spatially associated, it remains an interesting case worthy of further attention.