The melt rheology of two zinc alkali phosphate glasses was studied with oscillatory flow experiments to accelerate efforts to melt process them with organic polymers. The glasses differed markedly in chemical durability and melt viscosity. Melt viscosities were time independent at temperatures near their glass transition temperature. The ω dependence of the complex viscosity η* conformed excellently to predictions, η*(ω)=η0/(1+iωτ), for Hookean dumbbells. Fit parameters η0and τ were used to estimate a melt viscosity–average molecular weightMusing a model reported by Ferry and co‐workers [J. Applied Phys.26, 359 (1955)]. Arrhenius activation energies ΔH‡for viscous flow were consistent with those traditionally found for inorganic glasses. At higher temperatures, the melt viscosity of both glasses increased monotonically with time. The viscosity increase was exponential at long times, and the transition time to the exponential dependence was found to be strongly dependent on the shear strain. The steep viscosity rise is expected to cause significant processing challenges. Sample crystallization is thought to be responsible for this behavior.