A search for the non-resonant decays \(B^0_s \rightarrow \mu^{+}\mu^{-}\mu^{+}\mu^{-}\) and \(B^0 \rightarrow \mu^{+}\mu^{-}\mu^{+}\mu^{-}\) is presented. The measurement is performed using the full Run 1 data set collected in proton-proton collisions by the LHCb experiment at the LHC. The data correspond to integrated luminosities of \(1\) and \(2~\mathrm{fb}^{-1}\) collected at centre-of-mass energies of \(7\) and \(8~\mathrm{TeV}\), respectively. No signal is observed and upper limits on the branching fractions of the non-resonant decays at \(95\%\) confidence level are determined to be \begin{align*} \mathcal{B}(B^0_s \rightarrow \mu^{+}\mu^{-}\mu^{+}\mu^{-}) & < 2.5 \times 10^{-9} \mathcal{B}(B^0 \rightarrow \mu^{+}\mu^{-}\mu^{+}\mu^{-}) & < 6.9 \times 10^{-10}. \end{align*}