In order to investigate magnetic properties in the vicinity of the magnetic to superconducting (SC) phase boundary, we have performed Cu-NQR measurements under hydrostatic pressure (P) on two off-stoichiometric Ce1+xCu2+ySi2 polycrystals: Ce0.99Cu2.02Si2 (denoted as Ce 0.99) exhibits the anomalous ground state so-called A phase dominated by critical magnetic correlations and Ce0.975Cu2Si2 (Ce 0.975) shows a static magnetic ordering at ambient P. With increasing P which enhances the Kondo temperature, the fraction of the A phase becomes markedly suppressed, and the SC fraction increases correspondingly, although T-c stays nearly constant in the measured pressure range, P less than or equal to 14.4 kbar. This shows that the P-induced transition from the A to the SC phases is of a first-order type, without passing through a quantum (T -->0) phase transition. However, no superconductivity has been attained in Ce 0.975 up to 19.4 kbar. It is considered that the P-induced superconductivity is sensitive to the sample quality.