Accurate modeling of the time-dependent viscoelastic behavior of concrete, such as creep, is essential for reliable long-term structural performance prediction. While existing models are physically grounded, they often rely on simplifying assumptions, fixed functional, and extensive calibration. In this work, we investigate the potential of mechanics-informed machine learning (MIML) for constitutive modeling of concrete viscoelasticity. The training data are synthesized from a thermodynamically consistent viscoelastic model formulated using internal state variables derived from Helmholtz free energy and dissipation potentials. This synthetic dataset provides a reference for training and validating machine learning surrogates. In our MIML framework, the constitutive differential equations are embedded into the learning process, enforcing physical constraints and thermodynamic admissibility. This integration enables the network to learn stress-strain-time relationships while maintaining fundamental principles such as causality, non-negative dissipation, and time-dependent memory effects. The MIML model also functions as a surrogate time integrator for internal variable evolution and a predictor of total energy, enabling stable and efficient simulation of material response under varying loading histories. By capturing the evolution of internal variables, the model offers a thermodynamically consistent, reduced-order alternative to conventional numerical solvers. Compared to classical rheological formulations, MIML provides enhanced flexibility in representing complex, nonlinear, and age-dependent behavior without relying on closed-form expressions or complete history retention. It also supports real-time adaptability and efficient inference, making it suitable for applications such as digital twins and structural health monitoring. Numerical experiments demonstrate that MIML reproduces the viscoelastic response of concrete across diverse loading scenarios using limited synthetic data. This study underscores the potential of mechanics-informed machine learning as an interpretable and physics-aware approach to constitutive modeling in computational mechanics.