Summary: Consider an integral equation \(\lambda \; u - T u = f\), where \(T\) is an integral operator, defined on \(C[0, 1]\), with a kernel having an algebraic or a logarithmic singularity. Let \(\pi_m\) denote an interpolatory projection onto a space of piecewise polynomials of degree \(\le r - 1\) with respect to a graded partition of [0, 1] consisting of \(m\) subintervals. In the product integration method, an approximate solution is obtained by solving \(\lambda \; u_m - T \pi_m u_m = f\). As in order to achieve a desired accuracy, one may have to choose \(m\) large, we find approximations of \(u_m\) using a discrete modified projection method and its iterative version. We define a two-grid iteration scheme based on this method and show that it needs less number of iterates than the two-grid iteration scheme associated with the discrete collocation method. Numerical results are given which validate the theoretical results.