A search for the lepton-flavour violating decays B$_{s}^{0}$  → e$^{±}$μ$^{∓}$ and B$^{0}$ → e$^{±}$μ$^{∓}$ is performed based on a sample of proton-proton collision data corresponding to an integrated luminosity of 3 fb$^{−1}$, collected with the LHCb experiment at centre-of-mass energies of 7 and 8 TeV. The observed yields are consistent with the background-only hypothesis. Upper limits on the branching fraction of the B$_{s}^{0}$  → e$^{±}$μ$^{∓}$ decays are evaluated both in the hypotheses of an amplitude completely dominated by the heavy eigenstate and by the light eigenstate. The results are $ \frac{@}{@}\mathrm{\mathcal{B}}\left({B}_s^0\to {e}^{\pm }{\mu}^{\mp}\right)<6.3(5.4)\times 1{0}^{-9} $ and $ \frac{@}{@}\mathrm{\mathcal{B}}\left({B}_s^0\to {e}^{\pm }{\mu}^{\mp}\right)<7.2(6.0)\times 1{0}^{-9} $ at 95% (90%) confidence level, respectively. The upper limit on the branching fraction of the B$^{0}$ → e$^{±}$μ$^{∓}$ decay is also evaluated, obtaining $ \frac{@}{@}\mathrm{\mathcal{B}}\left({B}^0\to {e}^{\pm }{\mu}^{\mp}\right)<1.3(1.0)\times 1{0}^{-9} $ at 95% (90%) confidence level. These are the strongest limits on these decays to date.