We report the observation and study of the decay J/ψ→φηη′ using 1.3×109 J/ψ events collected with the BESIII detector. Its branching fraction, including all possible intermediate states, is measured to be (2.32±0.06±0.16)×10-4. We also report evidence for a structure, denoted as X, in the φη′ mass spectrum in the 2.0-2.1 GeV/c2 region. Using two decay modes of the η′ meson (γπ+π- and ηπ+π-), a simultaneous fit to the φη′ mass spectra is performed. Assuming the quantum numbers of the X to be JP=1-, its significance is found to be 4.4σ, with a mass and width of (2002.1±27.5±21.4) MeV/c2 and (129±17±9) MeV, respectively, and a product branching fraction B(J/ψ→ηX)×B(X→φη′)=(9.8±1.2±1.7)×10-5. Alternatively, assuming JP=1+, the significance is 3.8σ, with a mass and width of (2062.8±13.1±7.2) MeV/c2 and (177±36±35) MeV, respectively, and a product branching fraction B(J/ψ→ηX)×B(X→φη′)=(9.6±1.4±2.0)×10-5. The angular distribution of J/ψ→ηX is studied and the two JP 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.