  It is still not known whether a solution to the incompressible Euler
equation, endowed with a smooth initial value, can blow-up in finite time. In
[{\em Comm. Math. Phys.}, 378:557--568, 2020] it has been shown that, if it
exists, such a solution becomes linearly unstable close to the blow-up time. In
this paper, we show that the same phenomenon holds even in the more rigid
axisymmetric case. To obtain this result, we first prove a blow-up criterion
involving only the toroidal component of the vorticity. The instability of
blow-up profiles is also investigated.
