  We study the propagation of probe scalar fields in the background of 4D
Einstein-Gauss-Bonnet black holes with anti-de Sitter (AdS) asymptotics and
calculate the quasinormal modes. Mainly, we show that the quasinormal spectrum
consists of two different branches, a branch perturbative in the Gauss-Bonnet
coupling constant $\alpha$ and another branch nonperturbative in $\alpha$. The
perturbative branch consists of complex quasinormal frequencies that
approximate to the quasinormal frequencies of the Schwarzschild AdS black hole
in the limit of a null coupling constant. On the other hand, the
nonperturbative branch consists of purely imaginary frequencies and is
characterized by the growth of the imaginary part when $\alpha$ decreases,
diverging in the limit of null coupling constant, therefore they do not exist
for the Schwarzschild AdS black hole. Also, we find that the imaginary part of
the quasinormal frequencies is always negative for both branches; therefore,
the propagation of scalar fields is stable in this background.
