  We perform a complete one-loop computation of the two-body flavour-changing
top decays $t \rightarrow c h$ and $t \rightarrow c V$ ($V= \gamma, Z$), within
the aligned two-Higgs-doublet model. We evaluate the impact of the model
parameters on the associated branching ratios, taking into account constraints
from flavour data and measurements of the Higgs properties. Assuming that the
$125$~GeV Higgs corresponds to the lightest CP-even scalar of the CP-conserving
aligned two-Higgs-doublet model, we find that the rates for such
flavour-changing top decays lie below the expected sensitivity of the future
high-luminosity phase of the LHC. Measurements of the Higgs signal strength in
the di-photon channel are found to play an important role in limiting the size
of the $t \rightarrow c h$ decay rate when the charged scalar of the model is
light.
