  In this work, we have proposed a modular $A_4$ symmetric model of neutrino
mass which, simultaneously, explains observed baryon asymmetry of the
Universe(BAU). In minimal extension of the standard model(SM) with two
right-handed neutrinos we work in a supersymmetric framework. At Type-I seesaw
level, the model predicts scaling in the neutrino mass matrix. In order to have
correct low energy phenomenology, we propose two possible scenarios of
scale-breaking in the neutrino mass matrix emanating from Type-I seesaw.
Scenario-1 is based on the dimension-5 Weinberg operator whereas scenario-2
implements Type-II seesaw via scalar triplet Higgs
superfields($\Delta,\bar{\Delta}$). Interestingly, the breaking patterns in
both, otherwise dynamically different scenarios, are similar which can be
attributed to the same charge assignments of superfields($\Delta,\bar{\Delta}$)
and the Higgs superfield $H_u$ under modular $A_4$ symmetry. The breaking is
found to be proportional to the Yukawa coupling of modular weight
10($Y_{1,1'}^{10}$). We, further, investigates the predictions of the model
under scenario-2 (Type-I+II) for neutrino mass, mixings and matter-antimatter
asymmetry of the Universe. The model predicts normal hierarchical neutrino
masses and provide a robust range ($0.05-0.08$)eV for sum of neutrino
masses($\sum m_{i}$). Lepton number violating $0\nu\beta\beta$ decay
amplitude($M_{ee}$) is obtained to lie in the range ($0.04-0.06$)eV. Future
$0\nu\beta\beta$ decay experiments such as NEXT and nEXO shall pose crucial
test for the model. Both $CP$ conserving and $CP$ violating solutions are
allowed in the model. Interesting correlations are obtained, specially, between
Yukawa couplings of modular weight 2 and complex modulus $\tau$. The model
exhibit consistent explanation of BAU for right-handed Majorana neutrino mass
scale in the range ($(1-5)\times10^{13}$)GeV.
