A study of $B^0_s \to \eta_c \phi$ and $B^0_s \to \eta_c \pi^+ \pi^-$ decays is performed using $pp$ collision data corresponding to an integrated luminosity of 3.0 fb, collected with the LHCb detector in Run 1 of the LHC. The observation of the decay $B^0_s \to \eta_c \phi$ is reported, where the $\eta_c$ meson is reconstructed in the $p\bar p$, $K^+K^-\pi^+\pi^-$, $\pi^+\pi^-\pi^+\pi^-$ and $K^+K^-K^+K^-$ decay modes and the $\phi(1020)$ in the $K^+ K^-$ decay mode. The decay $B^0_s \to J/\psi \phi$ is used as a normalisation channel. Evidence is also reported for the decay $B^0_s \to \eta_c \pi^+\pi^-$, where the $\eta_c$ meson is reconstructed in the $p\bar p$ decay mode, using the decay $B^0_s \to J/\psi \pi^+ \pi^-$ as a normalisation channel. The measured branching fractions are \begin{eqnarray*} {\mathcal B (B^{0}_{s} \to \eta_{c} \phi)} &=& \left(5.01 \pm 0.53 \pm 0.27 \pm 0.63 \right) \times 10^{-4} \,, \nonumber \\ {\mathcal B (B^{0}_{s} \to \eta_{c} \pi^+ \pi^-)} &=& \left(1.76 \pm 0.59 \pm 0.12 \pm 0.29 \right) \times 10^{-4} \,, \end{eqnarray*} where in each case the first uncertainty is statistical, the second systematic and the third uncertainty is due to the limited knowledge of the external branching fractions.