Rigid rectangular footings resting on a semi-infinite elastic mass do not admit an exact analytical solution because the contact problem involves mixed boundary conditions acting on a non-smooth planform with corner singularities. While closed-form solutions exist for rigid circular, strip, and elliptical punches, rectangular footings are typically treated using numerical methods or empirical correction factors. This paper presents an approximate yet rational method for evaluating the contact pressure distribution and elastic settlement of rigid rectangular footings under uniform loading. The rectangular planform is first associated with an equivalent ellipse that preserves area and aspect ratio, allowing use of the classical pressure distribution of a rigid elliptical punch. A directional radial correction factor is then introduced to adapt the elliptical pressure field to the rectangular boundary. Elastic settlements are computed by convolving the mapped pressure distribution with the Green’s function of a finite elastic layer, implemented using image-term convolution, implemented efficiently using fast Fourier transform techniques. The methodology is verified by reproducing the exact settlement of rigid circular footings on a semi-infinite elastic mass and is demonstrated through a representative rectangular footing example. The proposed framework provides a transparent and computationally efficient alternative for estimating contact pressure and elastic settlement of rigid rectangular foundations. The formulation explicitly accounts for finite soil thickness and reduces to the classical half-space solution as a limiting case.