Nuclear fusion reactions at the sub-barrier energy regime (typically, 1 eV to a few keV) govern different fundamental aspects of the primordial nucleosynthesis in compact objects. One of the primary keys to understanding the relationship between stellar evolution and nuclear reaction dynamics is the energy dependence of astrophysical observables like fusion cross section $\sigma(E)$. We have studied $^3$He($\alpha$,$\gamma$)$^7$Be and $^3$He($^3$He,2p)$^4$He reactions by considering the quantum mechanical tunneling phenomenon. As an improvement over our earlier work, in this context, we invoked the idea of folding model potential for numerical computation of $\sigma(E)$. The results of our calculation are compared with those found in the literature. The present results are in good agreement with the experimental results.