<p>In this research study, we consider a multi-step version of Galerkin approaches, that is based on Jacobi polynomials and also powerful Gauss-Jacobi quadrature formulas, for solving certain classes of integro-differential models of the first order with non-vanishing delays, represented in difference form. Our basic motivation for proposing such this multi-step Galerkin approach is for the effect of difference delay in the considered models. Therefore, we can have a piecewise numerical solution in every sub-interval of the computational domain that is dependent on the value of the delay-difference parameter. Another motivation of our present research study is to directly prove the convergence analysis without considering any auxiliary problem, in which some quadrature rules were ignored in the discretization scheme of many research works in the case of global single-step Galerkin Jacobi approaches. It provides a better understanding of analysis of the errors at an exponential rate of convergence. We also provide some test problems and discretize them by the suggested scheme and we demonstrate the positive effect of p-refinement in our proposed approach experimentally to support the error analysis.</p>