In this paper, we analyze the conditions for convergence toward General Relativity of scalar-tensor gravity theories defined by an arbitrary coupling function $alpha$ (in the Einstein frame). We show that, in general, the evolution of the scalar field $(phi)$ is governed by two opposite mechanisms: an attraction mechanism which tends to drive scalar-tensor models toward Einsteins theory, and a repulsion mechanism which has the contrary effect. The attraction mechanism dominates the recent epochs of the universe evolution if, and only if, the scalar field and its derivative satisfy certain boundary conditions. Since these conditions for convergence toward general relativity depend on the particular scalar-tensor theory used to describe the universe evolution, the nucleosynthesis bounds on the present value of the coupling function, $alpha_0$, strongly differ from some theories to others. For example, in theories defined by $alpha propto midphimid$ analytical estimates lead to very stringent nucleosynthesis bounds on $alpha_0$ ($lesssim 10^{-19}$). By contrast, in scalar-tensor theories defined by $alpha propto phi$ much larger limits on $alpha_0$ ($lesssim 10^{-7}$) are found.
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