A recently proposed consistent approach for elastically scattering gas mixtures, of the type introduced by Bhatnagar, Gross, and Krook (BGK), has been extended to deal with a four species gas undergoing reversible bimolecular chemical reactions. The single BGK collision operator introduced for each species must take into account also transfer of mass and of energy of chemical bond. Suitable auxiliary fields have then to be introduced not only for temperatures and velocities, but also for densities, in order to fulfill correctly balance equations for mass, momentum, and total energy. The exact collision equilibrium, satisfying the mass action law of chemistry, is also recovered, and the proper choice of collision frequencies is discussed. Preliminary numerical results for the relaxation problem in space-homogeneous conditions are reported and briefly commented on.