Mechanical contact is a key phenomenon in the development and remodeling of biological tissues, where growth and morphogenesis can lead to evolving geometries and interactions between complex structures. However, classical Lagrangian finite-element methods often face significant challenges in such scenarios due to the need for sophisticated and computationally costly contact algorithms, especially when boundaries change as a result of growth. To address these limitations, we propose an Eulerian-based finite-element method that simplifies the modeling of mechanical contact~\cite{Lorez2024}, offering a promising avenue for applications in biomechanics and morphogenesis. Our approach employs a fixed-grid Eulerian formulation combined with a phase-field description of solid bodies, which naturally accommodates evolving geometries. The mechanical response of each solid is captured using the reference map technique~\cite{Kamrin2012}, while contact is handled through a volumetric penalty-based constraint~\cite{Lorez2024} that enforces non-interpenetration. This framework is particularly well-suited for problems involving growth-driven contact, where tracking moving boundaries and resolving interactions between multiple bodies are essential. While our current simulations focus on solid-solid contact scenarios, the framework is broadly applicable and naturally suited to problems involving growth, remodeling, and active material behavior. Its ability to robustly handle evolving geometries and mechanical interactions between complex-shaped bodies makes it a promising tool for future studies in developmental biomechanics and the mechanics of morphogenesis.