We report a measurement of the branching fraction for ψ(3770)→γχc1 and search for the transition ψ(3770)→γχc2 based on 2.92 fb−1 of e+e− data accumulated at s√=3.773 GeV with the BESIII detector at the BEPCII collider. We measure B(ψ(3770)→γχc1)=(2.48±0.15±0.23)×10−3, which is the most precise measurement to date. The upper limit on the branching fraction of ψ(3770)→γχc2 at a 90% confidence level is B(ψ(3770)→γχc2)<0.64×10−3. The corresponding partial widths are Γ(ψ(3770)→γχc1)=(67.5±4.1±6.7) keV and Γ(ψ(3770)→γχc2)<17.4 keV.