We consider the stochastic heat equation on a closed Riemannian manifold $M$ satisfying: \begin{equation*} \partial_tu(t,x)=\frac{1}{2}\Delta_Mu(t,x)+\sigma(t,x,u)\dot{W}(t,x),\quad (t,x)\in\mathbb{R}_+\times M, \end{equation*} where $\Delta_M$ denotes the Laplace-Beltrami operator, and $\dot{W}$ is a centered Gaussian noise that is white in time and colored in space. Assuming that $\sigma$ is Lipschitz in $u$ and uniformly bounded, we estimate small ball probabilities for the solution $u$ when $u(0,x)\equiv 0$.