Extrusion three-dimensional (3D) bioprinting typically requires an ad-hoc trial-and-error optimization of the bioink composition towards enhanced resolution. The bioink solutions are solidified after leaving cone-shaped or cylindrical nozzles. The presence of bioink instability not only hampers the extrusion resolution but also affects the behavior of embedded cellular components. This is a key factor in selecting bioinks and bioprinting design parameters for well-established desktop and handheld bioprinters. In this work, we developed an analytical solution for the process of bioink deposition and compared its predictions against numerical simulations of the deposition. We estimated the onset of bioink instability as a function of bioink rheological properties and nozzle geometry. Both analytical and simulation results demonstrated that enhancing shear-thinning behavior of the bioink stabilizes the printing process whereas bioink shear-thickening behavior induces an opposite effect through extending the toe region of the deposition. The present study serves as a benchmark for detailed simulations of the extrusion process for optimal bioprinting.