The centrality and system size dependence of multiplicity fluctuations of charged particles produced in nuclear collisions at 158A GeV was studied by the NA49 Collaboration. Centrality selected Pb+Pb collisions, semicentral C+C and Si+Si collisions as well as inelastic p+p interactions were analyzed. The number of projectile participants determined on an event-by-event basis was used to characterize the collision centrality. The scaled variance of the multiplicity distribution obtained in the forward rapidity region (1.1<yc.m.<2.6) shows a significant increase toward peripheral collisions. The results are similar for negatively and positively charged particles and about 50% larger for all charged particles. String-hadronic models of nuclear reactions without the fusion process do not reproduce the rise of fluctuations from central toward peripheral collisions. The measured centrality dependence can be reproduced in superposition models with the assumption of contributions from target participants to particle production in the forward hemisphere or in string models with fusion.