Using data samples collected at center-of-mass energies s√=4.23, 4.26, and 4.36 GeV with the BESIII detector operating at the BEPCII storage ring, we search for the production of the charmoniumlike state Y(4140) through a radiative transition followed by its decay to ϕJ/ψ. No significant signal is observed and upper limits on σ[e+e−→γY(4140)]⋅B(Y(4140)→ϕJ/ψ) at the 90% confidence level are estimated as 0.35, 0.28, and 0.33 pb at s√=4.23, 4.26, and 4.36 GeV, respectively.