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[en] In the framework of k_T -factorization approach, the production of prompt χ_c mesons in pp collisions at the LHC energies is studied. Our consideration is based on the off-shell amplitudes for hard partonic subprocesses g"*g"*→χ_c_J and non-relativistic QCD formalism for bound states. The transverse momentum dependent (unintegrated) gluon densities in a proton were derived from Ciafaloni-Catani-Fiorani-Marchesini evolution equation or, alternatively, were chosen in accordance with Kimber-Martin-Ryskin prescription. Taking into account both color singlet and color octet contributions, we deduce the corresponding non-perturbative long-distance matrix elements from the fits to the latest ATLAS data on χ_c_1 and χ_c_2 transverse momentum distributions at √(s)=7 TeV. We find that these distributions at small and moderate p_T are formed mainly by the color singlet components. We successfully described the data on the differential cross sections and relative production rates σ(χ_c_2)/σ(χ_c_1) presented by the ATLAS, CMS and LHCb Collaborations. We find that the fit points to unequal wave functions of χ_c_1 and χ_c_2 states.