Abstract A new type of cosmological transient, dubbed fast radio bursts (FRBs), was recently discovered. The source of FRBs is still unknown. One possible scenario of an FRB is the collapse of a spinning supra-massive neutron star. Zhang (2014, ApJ, 780, L21) suggests that the collapse can happen shortly (hundreds to thousands of seconds) after the birth of supra-massive neutron stars. The signatures can be visible in X-ray afterglows of long and short gamma-ray bursts (GRBs). For instance, a sudden drop (decay index steeper than $-3$ to $-9$) from a shallow decay (decay index shallower than $-1$) in the X-ray afterglow flux can indicate such an event. We selected the X-ray afterglow light curves with a steep decay after the shallow decay phase from the Swift/XRT GRB catalog. We analyzed when the decay index changed suddenly by fitting these light curves to double power-law functions and compared them with the onset of FRBs. We found that none of our GRB samples match the onset of FRBs.