The application of microfluidics technology in additive manufacturing is an emerging approach that makes possible the fabrication of functional three-dimensional cell-laden structured biomaterials. A key challenge that needs to be addressed using a microfluidic-based printhead (MBP) is increasing the controllability over the properties of the fabricated microtissue. Herein, an MBP platform is numerically simulated for the fabrication of solid and hollow microfibers using a microfluidic channel system with high level of controllability over the microfiber geometrical outcomes. Specifically, the generation of microfibers is enabled by studying the effects of microfluidic-based bioprinting parameters that capture the different range of design, bioink material, and process parameter dependencies as numerically modeled as a multiphysics problem. Furthermore, the numerical model is verified and validated, exhibiting good agreement with literature-derived experimental data in terms of microfiber geometrical outcomes. Additionally, a predictive mathematical formula that correlates the dimensionless process parameters with dimensionless geometrical outcomes is presented to calculate the geometrical outcomes of the microfibers. This formula is expected to be applicable for bioinks within a prescribed range of the density and viscosity value. The MBP applications are highlighted towards precision fabrication of heterogeneous microstructures with functionally graded properties to be used in organ generation, disease modeling, and drug testing studies.