Abstract A model for real space singlet pairs (RSP), hopping on the quasi 2D antiferromagnetic background of localized spins, is presented. It is found that the “gas parameter” of the RSP's reaches unity at low enough temperatures, however small the concentration of RSP's. This happens due to the local coupling of the RSP to the growing antiferromagnetic fluctuations in the 2D magnetic layers. The low temperature increase of the magnetic correlation length ξ is suppressed significantly by the low energy density fluctuations in the quasi 2D RSP system, which develop already far above the Bose condensation temperature T c. The effective “magnetic mass” of the RSP is found to be proportional to (In ξ/ a) 2, where a is the magnetic lattice spacing. Both the magnetic energy shift and the size of the “magnetic cloud” of the RSP are calculated. The results are applied to the explanation of the NMR behavior observed in the high- T c cuprates.