The extrusion-based bioprinting is an additive manufacturing (AM) technology enabling the production of biological constructs through the layer-by-layer deposition of a bio-ink. This latter is a cell-laden polymer-based aqueous solution featuring shear-thinning and viscoelastic behavior. When the bio-ink flows outside of the extruder, the extrudate swell phenomen occurs resulting in a loss of printing resolution. In order to describe such an effect, a free-surface problem shall be considered. Modeling flows with moving boundaries in the case under investigation is challenging since the complex interaction between capillarity and elasticity, potentially causing localized and significant alterations in surface curvature, as well as abrupt changes in the stress distribution within these regions. In this framework, the aim of the present work is to investigate the extrusion process of a bio-ink by means of finite element (FEM) formulation. The free-surface problem is described by the dimensionless incompressible and isothermal Cauchy equations coupled with the Giesekus differential constituive model. The Arbitrary Lagrangian-Eulerian (ALE) description is adopted to track the moving boundary alongside the numerical solution.