  We study the minimal extended seesaw mechanism with one sterile neutrino in a
3+1 framework using modular $S_3$ symmetry. The active-sterile neutrino models
are classified based on the assignments of $S_3$ representations and modular
weights of the left-handed lepton doublets, triplet right-handed neutrino, and
sterile neutrino. No scalar flavons are considered, and the flavor symmetry is
broken by the vacuum expectation value (vev) of the modulus $\tau$. For a
particular set of representations of the Leptons and Higgs field, we obtain
eleven (11) different models based on different modular weights of charged
lepton $(k_L)$ and right-handed neutrino ($k_N$). Out of these, we consider two
models, which are discriminated by carrying out the numerical analysis so that
the parameter space in each model can fit the latest neutrino oscillation data
at 3$\sigma$. The Planck cosmological bound on the upper limit of the sum of
the active neutrino masses $\sum m_i <0.12$eV is also considered. Finally, the
best-fit parameters of the neutrino observables and model predictions are
evaluated using the minimum $\chi^2$ analysis.
