  Extragalactic fast X-ray transients (FXTs) are short-duration ($\sim$ ks)
X-ray flashes of unknown origin, potentially arising from binary neutron star
mergers. We investigate the possible link between FXTs and the afterglows of
off-axis merger-induced gamma-ray bursts (GRBs). By modelling the broadband
afterglows of 13 GRBs, we make predictions for their X-ray light curve
behaviour had they been observed off-axis, considering both a uniform jet with
core angle $\theta_{C}$ and a Gaussian-structured jet with truncation angle
$\theta_{W} = 2\theta_{C}$. We compare their peak X-ray luminosity, duration,
and temporal indices with those of the currently known extragalactic FXTs. Our
analysis reveals that a slightly off-axis observing angle of
$\theta_{\text{obs}}\approx (2.2-3)\theta_{C}$ and a structured jet are
required to explain the shallow temporal indices of the FXT light curves. In
the case of a structured jet, the durations of the FXTs are consistent with
those of the off-axis afterglows for the same range of observing angles,
$\theta_{\text{obs}}\approx (2.2-3)\theta_{C}$. While the off-axis peak X-ray
luminosities are consistent only for $\theta_{\text{obs}} = 2.2\theta_{C}$,
focussing on individual events reveals that the match of all three properties
of the FXTs at the same viewing angle is possible in the range
$\theta_{\text{obs}} \sim (2.2-2.6)\theta_{C}$. Despite the small sample of
GRBs analysed, these results show that there is a region of the parameter space
- although quite limited - where the observational properties of off-axis GRB
afterglows can be consistent with those of the newly discovered FXTs. Future
observations of FXTs discovered by the recently launched Einstein Probe mission
and GRB population studies combined with more complex afterglow models will
shed light on this possible GRB-FXT connection, and eventually unveil the
progenitors of some FXTs.
