  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 Type-I+II seesaw scenario in a supersymmetric
framework. At Type-I seesaw level, the model predicts scaling in the neutrino
mass matrix. We implement Type-II seesaw to break the scaling and to have
correct low energy phenomenology. The breaking is proportional to the $vev$
alignment of the scalar triplet implementing Type-II seesaw and Yukawa coupling
of modular weight 6($Y_1^{6}$). The model predicts normal hierarchical neutrino
masses and provide a robust range ($0.05-0.07$)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 $\tau$. Contrary to
$Y_2^{2}$ and $Y_3^{2}$, the Yukawa coupling $Y_1^{2}$ is found to be
insensitive to $\tau$ and thus to $CP$ violation because complex modulus $\tau$
is the only source of $CP$ violation in the model. We, also, investigate the
prediction of the model for BAU. The model exhibit consistent explanation of
BAU for right-handed Majorana neutrino mass scale in the range
($10^{11}-10^{13}$)GeV.
