The deformation and cross-streamline migration of an elastic particle in pressure-driven flows of Newtonian and viscoelastic (Oldroyd-B, Giesekus) fluids in a cylindrical tube are studied through 3D finite element method numerical simulations. The dependences of particle deformation and migration on geometric confinement, flow strength, and fluid rheology are investigated. If the particle is initially not at the channel axis, it attains an asymmetric shape and migrates. In a Newtonian liquid, the migration is always directed towards the tube axis. A project equation is proposed for the design of a microfluidic cylindrical device aimed at focusing elastic particles on the cylinder centerline. In a viscoelastic liquid, the migration direction and velocity depend on the competition among particle deformability, fluid elasticity, and fluid viscosity shear thinning (if any). In a certain range of parameters, an unstable radial position appears, which separates the region where the migration is directed towards the axis from the region where it is directed towards the wall.