We investigate the enrichment in elements produced by the slow neutron-capture process (s-process) in the globular clusters M4 (NGC 6121) and M22 (NGC 6656). Stars in M4 have homogeneous abundances of Fe and neutron-capture elements, but the entire cluster is enhanced in s-process elements (Sr, Y, Ba, Pb) relative to other clusters with a similar metallicity. In M22, two stellar groups exhibit different abundances of Fe and s-process elements. By subtracting the mean abundances of s-poor from s-rich stars, we derive s-process residuals or empirical s-process distributions for M4 and M22. We find that the s-process distribution in M22 is more weighted toward the heavy s-peak (Ba, La, Ce) and Pb than M4, which has been enriched mostly with light s-peak elements (Sr, Y, Zr). We construct simple chemical evolution models using yields from massive star models that include rotation, which dramatically increases s-process production at low metallicity. We show that our massive star models with rotation rates of up to 50% of the critical (break-up) velocity and changes to the preferred 17O(α, γ)21Ne rate produce insufficient heavy s-elements and Pb to match the empirical distributions. For models that incorporate asymptotic giant branch yields, we find that intermediate-mass yields (with a 22Ne neutron source) alone do not reproduce the light-to-heavy s-element ratios for M4 and M22, and that a small contribution from models with a 13C pocket is required. With our assumption that 13C pockets form for initial masses below a transition range between 3.0 and 3.5 M ☉, we match the light-to-heavy s-element ratio in the s-process residual of M22 and predict a minimum enrichment timescale of between 240 and 360 Myr. Our predicted value is consistent with the 300 Myr upper limit age difference between the two groups derived from isochrone fitting.