We discuss the self-enrichment scenario by asymptotic giant branch (AGB) stars for the formation of multiple populations in globular clusters (GCs) by analysing data set of giant stars observed in nine Galactic GCs, covering a wide range of metallicities and for which the simultaneous measurements of C, N, O, Mg, Al, and Si are available. To this aim, we calculated six sets of AGB models, with the same chemical composition as the stars belonging to the first generation of each GC. We find that the AGB yields can reproduce the set of observations available, not only in terms of the degree of contamination shown by stars in each GC but, more important, also the observed trend with metallicity, which agrees well with the predictions from AGB evolution modelling. While further observational evidences are required to definitively fix the main actors in the pollution of the interstellar medium from which new generation of stars formed in GCs, the present results confirm that the gas ejected by stars of mass in the range 4 M_{⊙} ≤ M ≤ 8 M_{⊙} during the AGB phase share the same chemical patterns traced by stars in GCs.