When contact between two solids is modeled, the Lagrangian finite-element method is generally considered to be the way to go. However, most modern contact methods, which have been extensively developed over the past few decades, are relatively complex and computationally costly with complex algorithms needed for contact detection and resolution. This complexity increases even further when the problems of interest involve evolving boundaries due to material growth. Such situations may occur for instance for crystallization phenomena in porous media. Here, we present an alternative approach that overcomes many of the associated challenges by using an Eulerian finite-element approach rather than the traditional Lagrangian approach to solve for contact between elastic solids [1]. Our approach is based on an Eulerian framework with a fixed mesh and incorporates a phase-field method that provides a diffuse representation of the solid bodies. This significantly simplifies the model for surface evolution. The elastic response of the solid is modeled using the reference map technique [2]. Additionally, the methodology introduces a volumetric contact constraint using a penalty-based method to solve contact and prevent interpenetration of solids. We present various numerical examples to demonstrate the validity and versatility of our proposed method and its ability to accurately solve solid-solid interactions of complex-shaped bodies. Most importantly, the proposed Eulerian-based approach with the phase-field formulation considerably streamlines the contact detection and resolution processes.