  Texture zeros in fermion mass matrices have been widely considered in
tackling the Standard Model flavour puzzle. In this work, we perform a
systematic analysis of texture zeros in lepton mass matrices in the framework
of $\Gamma_{3}'\cong T'$ modular symmetry. Assuming that the lepton fields
transform as irreducible representations of $T'$, we obtain all possible
texture-zero patterns for both charged-lepton and neutrino mass matrices which
can be achieved from $T'$ modular symmetry. We provide representative models
for the phenomenologically-viable textures which can accommodate the
experimental data. The predictions for lepton mixing angles, CP-violating
phases, light neutrino masses and effective neutrino mass relevant for
neutrinoless doble beta decay, are discussed. We find that the minimal viable
lepton model depends on only $7$ real free parameters including the modulus
$\tau$ (the corresponding charged-lepton mass matrix contains $4$ vanishing
entries, and the neutrino mass matrix has $1$ texture zero). Finally, we study
in detail three benchmark models, one for each neutrino mass generation
mechanism considered (Dirac, Majorana via Weinberg operator and Majorana via
minimal type-I seesaw mechanism).
