Approximation via multi-component reaction-diffusion system is an approximation method for a given partial differential equation by using a multicomponent reaction-diffusion system. In this article, we consider the initial value problem for reaction-diffusion equations with a general convolution integral which has recently been used in biology. We obtain the existence, uniqueness, and uniform boundedness of a solution for this problem and later propose a multi-component reaction-diffusion system that approximates it. To approximate the general convolution integral, we use solutions of heat equations with an advection term and an inhomogeneous term. Using such an approximation, we prove that the solution of the reaction-diffusion equation with the general convolution integral converges to the first component of the solution to the multi-component reaction-diffusion system.