Summary: The aim of this article is to establish the existence of the solution of non-linear functional integral equations \(x(l,h)=(U(l,h,x(l,h))+F (l,h,\int_0^l \int_0^h P (l, h, r, u, x(r, u))drdu,x,(l, h)))\times G (l,h,\int_0^a \int_0^a Q (l,h,r,u,x,(r,u))drdu, x(l,h))\) of two variables, which is of the form of two operators in the setting of Banach algebra \(C([0,a]\times[0,a])\), \(a>0\). Our methodology relies upon the measure of noncompactness related to the fixed point hypothesis. We have used the measure of noncompactness on \(C([0,a]\times[0,a])\) and a fixed point theorem, which is a generalization of Darbo’s fixed point theorem for the product of operators. We additionally illustrate our outcome with the help of an interesting example.