We propose and demonstrate single fiber dual-functionality optical tweezers based on a graded-index multimode fiber. By using the multi-angle fiber grinding and polishing technology, we fabricate the multimode fiber tip to be a special tapered shape, contributing to focus the outgoing beam with a large intensity gradient for the first functionality—three-dimensional contactless trapping of a microparticle. By adjusting the radial direction offset between the lead-in single mode fiber and the graded-index multimode fiber, we perform the second functionality—axial shift of the trapped microparticle with respect to the fiber tip without need of moving the fiber probe itself. It is convenient for practical applications. The theoretical and experimental results about the relationship between the radial offset and the equilibrium positions of the microparticle have the good consistency. Tailoring the trap and axial shift of the microparticle based on the graded-index multimode fiber provides convenient avenues for fiber optical tweezers applied in practical researches.