We study the production of forward di-jets in dilute-dense hadronic collisions, such as proton-lead collisions at the LHC. These final-state configurations, with both jets produced in the forward direction, probe the gluon density of the lead nucleus at very small longitudinal momentum fractions, and may therefore be sensitive to the non-linear ``saturation'' regime of QCD. Starting from the Color Glass Condensate framework, we are able to obtain a factorization formula for this process, that involves several transverse-momentum-dependent (TMD) gluons distributions for the nucleus, as opposed to a single collinear gluon distribution in the linear regime. We identify possible nuclear modifications of the structure of forward di-jets in proton-nucleus collisions versus proton-proton collisions, that would signal the necessity of including these saturation effects in QCD calculations.