Nuclear-reaction calculations of the heavy-element synthesis occurring in thermally pulsing asymptotic giant branch (AGB) stars were performed to analyze the observations of enhanced s-process elements in barium stars. In addition, s-process abundances were obtained from single neutron exposure calculations, and significant differences were found in the calculated abundances from previously published data. It was found that a mass transfer mechanism from an AGB star, in which the Ne-22 source operates, cannot be responsible for the s-process abundances observed in the classical barium stars HR 774 and Zeta Cap, and in the mild barium star Omicron Vir. A C-13 neutron source AGB star could better reproduce the abundance data of these stars. However, a potent C-13 neutron source has so far only been predicted to be active in low-metallicity AGB stars, and a low metallicity is not compatible with the population characteristics of the barium stars. Future observations needed to reconcile these findings are discussed. 81 references.