  We report the observation and study of the decay
$J/\psi\rightarrow\phi\eta\eta'$ using $1.3\times{10^9}$ $J/\psi$ events
collected with the BESIII detector. Its branching fraction, including all
possible intermediate states, is measured to be
$(2.32\pm0.06\pm0.16)\times{10^{-4}}$. We also report evidence for a structure,
denoted as $X$, in the $\phi\eta'$ mass spectrum in the $2.0-2.1$ GeV/$c^2$
region. Using two decay modes of the $\eta'$ meson ($\gamma\pi^+\pi^-$ and
$\eta\pi^+\pi^-$), a simultaneous fit to the $\phi\eta'$ mass spectra is
performed. Assuming the quantum numbers of the $X$ to be $J^P = 1^-$, its
significance is found to be 4.4$\sigma$, with a mass and width of $(2002.1 \pm
27.5 \pm 21.4)$ MeV/$c^2$ and $(129 \pm 17 \pm 9)$ MeV, respectively, and a
product branching fraction
$\mathcal{B}(J/\psi\rightarrow\eta{}X)\times{}\mathcal{B}(X\rightarrow\phi\eta')=(9.8
\pm 1.2 \pm 1.7)\times10^{-5}$. Alternatively, assuming $J^P = 1^+$, the
significance is 3.8$\sigma$, with a mass and width of $(2062.8 \pm 13.1 \pm
7.2)$ MeV/$c^2$ and $(177 \pm 36 \pm 35)$ MeV, respectively, and a product
branching fraction
$\mathcal{B}(J/\psi\rightarrow\eta{}X)\times{}\mathcal{B}(X\rightarrow\phi\eta')=(9.6
\pm 1.4 \pm 2.0)\times10^{-5}$. The angular distribution of
$J/\psi\rightarrow\eta{}X$ is studied and the two $J^P$ assumptions of the $X$
cannot be clearly distinguished due to the limited statistics. In all
measurements the first uncertainties are statistical and the second systematic.
