When fiber suspensions settle down in a headbox of a paper machine by gravity, the suspensions may create a weak vertical shear flow in the vicinity of the suspensions due to the variation of fiber number density. A fiber near the suspension may experience a shear force due to the vertical stream lines and migrate laterally either toward the high particle number density region or toward the lower particle density region depending on the settling Reynolds number. Such the cross-stream or lateral migration of a single fiber settling in a bounded weak shear flow with vertical streamlines, produced by a perturbation to the fiber number density, is studied using lattice Boltzmann simulations. The present simulation results show that at a given shear rate, the lateral migration can be divided into three phases depending on settling Reynolds number Rsd and fiber aspect ratio κ. When κ is given, at a low settling Reynolds number Rsd the shear force may dominate the fiber rotational inertia and the fiber will tumble and migrate toward the lower fiber density region and the fiber is finally stopped in the cross-stream direction by the wall repulsive force. This is phase one where the suspension becomes more stable. As Rsd increases and exceeds a critical settling Reynolds number Rsd1, the fiber attempts to align its long body with the horizontal direction due to inertial torque. On the other hand, the torque due to the imposed weak vertical shear flow rotates the non-spherical fiber in the opposite direction. The dynamic balance between the two torques may lead to a small angle between the fiber long body and horizontal plane, which may drive fiber migrate toward the higher fiber density region. This is phase two where the suspension is destabilized. This mechanism recently proposed by Shin, Koch and Subramanian 2009 is examined and confirmed by the present simulation results. As the settling Reynolds number continuously increases and exceeds a second critical Rsd2, the orientational inertia dominates the imposed vertical shear force and the fiber will remain in the center between two wall on time average. This is phase three.