  We consider flat Friedmann-Lema\^{\i}tre-Robertson-Walker cosmological models
in the framework of general scalar-tensor theories of gravity with arbitrary
coupling functions, set in the Jordan frame, in the cosmological epoch when the
energy density of the ordinary dust matter dominates over the energy density of
the scalar potential. Motivated by cosmological observations, we apply an
approximation scheme in the regime close to the so-called limit of general
relativity. The ensuing nonlinear approximate equations for the scalar field
and the Hubble parameter can be solved analytically in cosmological time. This
allows us to distinguish the theories with solutions that asymptotically
converge to general relativity and draw some implications about the
cosmological dynamics near this limit.
