Evaluating the Newton potential is crucial for efficiently solving the boundary integral equation of the Dirichlet boundary value problem of the Poisson equation. In the context of the Fourier–Garlerkin method for solving the boundary integral equation, we propose a fast algorithm for evaluating Fourier coefficients of the Newton potential by using a sparse grid approximation. When the forcing function of the Poisson equation expressed in the polar coordinates has $m$th-order bounded mixed derivatives, the proposed algorithm achieves an accuracy of order $\mathcal{𝒪}\left(n^{-m}{log}^{3}n\right)$, with requiring $\mathcal{𝒪}\left(n{log}^{2}n\right)$ number of arithmetics for the computation, where $n$ is the number of quadrature points used in one coordinate direction. With the help of this algorithm, the boundary integral equation derived from the Poisson equation can be efficiently solved by a fast fully discrete Fourier–Garlerkin method.