  A search for the lepton-flavour violating decays $B^0_s \rightarrow e^\pm
\mu^\mp$ and $B^0 \rightarrow e^\pm \mu^\mp$ 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 fractions are determined to be
$\mathcal{B}(B^0_s \rightarrow e^\pm \mu^\mp) < 5.4\,(6.3) \times 10^{-9}$ and
$\mathcal{B}(B^0 \rightarrow e^\pm \mu^\mp) < 1.0\,(1.3) \times 10^{-9}$ at
$90\%\,(95\%)$ confidence level, which are the strongest limits on these decays
to date.
