A search for the very rare $B^{*0}\rightarrow \mu ^+\mu ^-$ and $B_{s}^{*0}\rightarrow \mu ^+\mu ^-$ decays is conducted by analysing the $B_c^+\rightarrow \pi ^+\mu ^+\mu ^-$ process. The analysis uses proton-proton collision data collected with the LHCb detector between 2011 and 2018, corresponding to an integrated luminosity of 9 $\text {,fb}^{-1}$ . The signal signatures correspond to simultaneous peaks in the $\mu ^+\mu ^-$ and $\pi ^+\mu ^+\mu ^-$ invariant masses. No evidence for an excess of events over background is observed for either signal decay mode. Upper limits at the $90%$ confidence level are set on the branching fractions relative to that for $B_c^+\rightarrow J\hspace{-1.66656pt}/\hspace{-1.111pt}\psi \pi ^+$ decays, $\begin{aligned} \mathcal{R}{B^{*0}(\mu ^+\mu ^-)\pi ^+/J\hspace{-1.66656pt}/\hspace{-1.111pt}\psi \pi ^+}&< 3.8\times 10^{-5}\ \text { and }\ \mathcal{R}{B_{s}^{*0}(\mu ^+\mu ^-)\pi ^+/J\hspace{-1.66656pt}/\hspace{-1.111pt}\psi \pi ^+}&< 5.0\times 10^{-5}. \end{aligned}$ R B ∗ 0 ( μ + μ - ) π + / J / ψ π + < 3.8 × 10 - 5 and R B s ∗ 0 ( μ + μ - ) π + / J / ψ π + < 5.0 × 10 - 5 .