  We investigate the possibility to describe the quark mass hierarchies as well
as the CKM quark mixing matrix without fine-tuning in a quark flavour model
with modular $A_4$ symmetry. The quark mass hierarchies are considered in the
vicinity of the fixed point $\tau = \omega \equiv \exp({i\,2\pi/3})$ (the left
cusp of the fundamental domain of the modular group), $\tau$ being the VEV of
the modulus. The model involves modular forms of level 3 and weights 6, 4 and
2, and contains eight constants, only two of which, $g_u$ and $g_d$, can be a
source of CP violation in addition to the VEV of the modulus, $\tau = \omega +
\epsilon$, $(\epsilon)^* \neq \epsilon$, $|\epsilon|\ll 1$. We find that in the
case of real (CP-conserving) $g_u$ and $g_d$ and common $\tau$ ($\epsilon$) in
the down-quark and up-quark sectors, the down-type quark mass hierarchies can
be reproduced without fine tuning with $|\epsilon| \cong 0.03$, all other
constants being of ${\cal O}(1)$, and correspond approximately to $1 :
|\epsilon| : |\epsilon|^2$. The up-type quark mass hierarchies can be achieved
with the same $|\epsilon| \cong 0.03$ but allowing $g_u\sim {\cal O}(10)$ and
correspond to $1 : |\epsilon|/|g_u| : |\epsilon|^2/|g_u|^2$. In this setting,
we discuss the CKM quark mixing and CP violation.
