  As is known from previous studies the lepton number violating decays $K^+\to
\pi^- + l^{+}_1 + l^{+}_2$ have good prospects to probe new physics beyond the
Standard Model and provide valuable information on neutrino masses and mixing.
  We analyze these processes with an emphasis on their hadronic structure
aspects applying relativistic constituent quark model. We conclude that the
previously ignored contribution associated with the t-channel Majorana neutrino
exchange is comparable with the s-channel one in a wide range of neutrino
masses. We also estimated model independent absolute upper bounds on neutrino
contribution to these decays.
