  The rare flavour changing top quark decay $t\to cZ$ is a clear sign of new
physics and experimentally very interesting due to the huge number of top
quarks produced at the LHC. However, there are few (viable) models which can
generate a sizable branching ratio for $t\to cZ$ -- in fact vector-like quarks
seem to be the only realistic option. In this paper, we investigate all three
representations (under the Standard Model gauge group) of vector-like quarks
($U$, $Q_1$ and $Q_7$) that can generate a sizable branching ratio for $t\to
cZ$ without violating bounds from $B$ physics. Importantly, these are exactly
the three vector-like quarks which can lead to a sizable positive shift in the
prediction for $W$ mass, via the couplings to the top quark also needed for a
sizable Br($t\to cZ$). Calculating and using the one-loop matching of
vector-like quarks on the Standard Model Effective Field Theory, we find that
Br($t\to cZ$) can be of the order of $10^{-6}$, $10^{-5}$ and $10^{-4}$ for
$U$, $Q_1$ and $Q_7$, respectively and that in all three cases the large $W$
mass measurement can be accommodated.
