This paper presents a numerical expansion-iterative method for solving linear Volterra and Fredholm integral equations of the second kind. The method is based on vector forms of block-pulse functions and their operational matrix. By using this approach, solving the second kind integral equation reduces to solve a recurrence relation. The approximate solution is most easily produced iteratively via the recurrence relation. Therefore, computing the numerical solution does not need to directly solve any linear system of algebraic equations and to use any matrix inversion. Moreover, this approach does not use any projection method such as collocation, Galerkin, etc., for setting up the recurrence relation. To show convergence and stability of the method, some computable error bounds are obtained, and some test problems are provided to illustrate its accuracy and computational efficiency.