  We compute the one-loop contribution to the $\bar{\theta}$-parameter of an
axion-like particle (ALP) with CP-odd derivative couplings. Its contribution to
the neutron electric dipole moment is shown to be orders of magnitude larger
than that stemming from the one-loop ALP contributions to the up- and
down-quark electric and chromoelectric dipole moments. This strongly improves
existing bounds on ALP-fermion CP-odd interactions, and also sets limits on
previously unconstrained couplings. The case of a general singlet scalar is
analyzed as well. In addition, we explore how the bounds are modified in the
presence of a Peccei-Quinn symmetry.
