The production of J/psi mesons in jets is studied in the forward region of proton-proton collisions using data collected with the LHCb detector at a center-of-mass energy of 13 TeV. The fraction of the jet transverse momentum carried by the J/psi meson, z(J/psi) p(T) (J/psi) / p(T)(jet) is measured using jets with p(T)(jet) > 20 GeV in the pseudorapidity range 2.5 < eta(jet) < 4.0. The observed z(J/psi) distribution for J/psi mesons produced in b-hadron decays is consistent with expectations. However, the results for prompt J/psi production do not agree with predictions based on fixed-order nonrelativistic QCD. This is the first measurement of the p(T) fraction carried by prompt J/psi mesons in jets at any experiment.