  Evidence for the decays $B^{0}\rightarrow\bar{D}^{0}\phi$ and $B^{0}
\rightarrow \bar{D}^{*0}\phi$ is reported with a significance of 3.6$\,\sigma$
and 4.3$\,\sigma$, respectively. The analysis employs $pp$ collision data at
centre-of-mass energies $\sqrt{s}=7$, 8 and 13 TeV collected by the LHCb
detector and corresponding to an integrated luminosity of 9 $\rm{fb}^{-1}$. The
branching fractions are measured to be
$\mathcal{B}(B^{0}\rightarrow\bar{D}^{0}\phi) =
(7.7\pm2.1\pm0.7\pm0.7)\times10^{-7}$,$\mathcal{B}(B^{0} \rightarrow
\bar{D}^{*0}\phi)=(2.2\pm0.5\pm0.2\pm0.2)\times10^{-6}$. In these results, the
first uncertainties are statistical, the second systematic, and the third is
related to the branching fraction of the
$B^{0}\rightarrow\bar{D}^{0}K^{+}K^{-}$ decay, used for normalisation. By
combining the branching fractions of the decays
$B^{0}\rightarrow\bar{D}^{(*)0}\phi$ and
$B^{0}\rightarrow\bar{D}^{(*)0}\omega$, the $\omega$-$\phi$ mixing angle
$\delta$ is constrained to be $\tan^2\delta = (3.6\pm0.7\pm0.4)\times10^{-3}$,
where the first uncertainty is statistical and the second systematic. An
updated measurement of the branching fractions of the $B^{0}_{s} \rightarrow
\bar{D}^{(*)0}\phi$ decays, which can be used to determine the CKM angle
$\gamma$, leads to
$\mathcal{B}(B^{0}_{s}\rightarrow\bar{D}^{0}\phi)=(2.30\pm0.10 \pm
0.11\pm0.20)\times10^{-5}$,$\mathcal{B}(B^{0}_{s}\rightarrow\bar{D}^{*0}\phi)
=(3.17\pm0.16 \pm 0.17 \pm 0.27)\times10^{-5}$.
