In this paper, we study the dynamics of deformable cells in a channel flow of Newtonian and polymeric fluids and unravel the effects of deformability, elasticity, inertia, and size on the cell motion. We investigate the role of polymeric fluids on the cell migration behavior and the performance of inertial microfluidic devices. Our results show that the equilibrium position of the cell is on the channel diagonal, in contrast to that of rigid particles, which is on the center of the channel faces for the same range of Reynolds number. A constant-viscosity polymeric fluid, modeled using an Oldroyd-B constitutive equation, drives the cells toward the channel centerline, while a shear-thinning polymeric fluid, modeled using a Giesekus constitutive equation, pushes the cells toward the channel wall. The findings of this paper suggest that the addition of polymers in microfluidic devices can be used to enhance the throughput of cell focusing and separation devices at a low cost. This study provides an insight on the role of rheological properties of the fluid and the ways that they can be tuned to control the focal position of the cells.