We study the production at the LHC of a forward J/ψ meson and a backward jet with a large rapidity separation using the BFKL formalism. We give predictions for both the Non Relativistic QCD (NRQCD) approach to charmonium production and the Color Evaporation Model. In NRQCD, we find that the 3S18 part of the onium wavefunction is completely dominating the process, which makes the presented study a good probe for this color octet contribution.