This paper deals with the homogenization of a quasilinear elliptic problem having a singular lower order term and posed in a two-component domain with an $\epsilon$-periodic imperfect interface. We prescribe a Dirichlet condition on the exterior boundary, while we assume that the continuous heat flux is proportional to the jump of the solution on the interface via a function of order $\epsilon^\gamma$.\\ We prove an homogenization result for $-1<\gamma<1$ by means of the periodic unfolding method, adapted to two-component domains by P. Donato, K.H. Le Nguyen and R. Tardieu.\\ One of the main tools in the homogenization process is the study of a suitable auxiliary linear problem and a related convergence result. It shows that the gradient of $u^{\epsilon}$ behaves like that of the solution of the auxiliary one, associated with a weak cluster point of the sequence $\{u^{\epsilon}\}$, as $\epsilon \to 0$. This allows us not only to pass to the limit in the quasilinear term, but also to study the singular term near its singularity, via an accurate a priori estimate.