  The $B_s^0\pi^\pm$ invariant mass distribution is investigated in order to
search for possible exotic meson states. The analysis is based on a data sample
recorded with the LHCb detector corresponding to $3$ fb$^{-1}$ of $pp$
collision data at $\sqrt{s} = 7$ and $8$ TeV. No significant excess is found,
and upper limits are set on the production rate of the claimed $X(5568)$ state
within the LHCb acceptance. Upper limits are also set as a function of the mass
and width of a possible exotic meson decaying to the $B_s^0\pi^\pm$ final
state. The same limits also apply to a possible exotic meson decaying through
the chain $B_s^{*0}\pi^\pm$, $B_s^{*0} \to B_s^0 \gamma$ where the photon is
excluded from the reconstructed decays.
