The migration of ellipsoidal particles in bounded shear flow of Giesekus fluids is studied numerically using the direct forcing/fictitious domain method for the Weissenberg number ranging from 0.1 to 3.0, the mobility parameter α which quantifies the shear-thinning effect ranging from 0.1 to 0.7. The model and numerical method are validated by comparing the present results with available theoretical and numerical results in other literatures. The results show that the trajectory of particles depends on their initial orientation and vertical position, and the particle migration can be roughly classified into returning and passing pattern. The values of initial vertical position of particle corresponding to the separatrix between the returning and passing pattern decrease with increasing Weissenberg number regardless of the initial orientation of particle, and the shear thinning has the opposite effect. The evolution of particle orientation depends on the initial particle orientation. For the particles whose initial orientation is parallel to the shear plane, the particle rotates with the semi-major axis as radius in the shear plane. For the particles whose initial orientation is perpendicular to the shear plane, the particle rotates with the semi-minor axis as radius. For the particles whose initial orientation has a certain angle with the shear plane, the particle rotates with the vorticity axis and the orientation vector is gradually close to the vorticity vector. The evolution of the particle orientation becomes slow with increasing Wi whether it is in passing behavior or in returning behavior.