We investigate the nucleon-nucleon interaction by using the meson exchange model and the two-body Dirac equations of constraint dynamics. This approach to the two-body problem has been successfully tested for QED and QCD relativistic bound states. An important question we wish to address is whether or not the two-body nucleon-nucleon scattering problem can be reasonably described in this approach as well. This test involves a number of related problems. First we must reduce our two-body Dirac equations exactly to a Schrödinger-like equation in such a way that allows us to use techniques to solve them already developed for Schrödinger-like systems in nonrelativistic quantum mechanics. Related to this, we present a new derivation of Calogero’s variable phase shift differential equation for coupled Schrödinger-like equations. Then we determine if the use of nine meson exchanges in our equations gives a reasonable fit to the experimental scattering phase shifts for $n-p$ scattering. The data involve seven angular momentum states including the singlet states ${}^1{S}_0,$ ${}^1{P}_1,$ ${}^1{D}_2$ and the triplet states ${}^3{P}_0,$ ${}^3{P}_1,$ ${}^3{S}_1,$ ${}^3{D}_1.$ Two models that we have tested give us a fairly good fit. The parameters obtained by fitting the $n-p$ experimental scattering phase shift give a fairly good prediction for most of the $p-p$ experimental scattering phase shifts examined (for the singlet states ${}^1{S}_0,$ ${}^1{D}_2$ and triplet states ${}^3{P}_0,$ ${}^3{P}_1).\mathrm{}$ Thus the two-body Dirac equations of constraint dynamics present us with a fit that encourages the exploration of a more realistic model. We outline generalizations of the meson exchange model for invariant potentials that may possibly improve the fit.