Trapping and manipulating mesoscopic biological cells with high precision and flexibility are very important for numerous biomedical applications. In particular, a photonic nanojet based on a non-resonance focusing phenomenon can serve as a powerful tool for manipulating red blood cells and tumor cells in blood. In this study, we demonstrate an approach to trap and drive cells using a high-quality photonic nanojet which is produced by a specific microcone-shaped optical-fiber tip. The dynamic chemical etching method is used to fabricate optical-fiber probes with a microcone-shaped tip. Optical forces and potentials exerted on a red blood cell by a microcone-shaped fiber tips are analyzed based on finite-difference time-domain calculations. Optical trapping and driving experiments are done using breast cancer cells and red blood cells. Furthermore, a cell chain is formed by adjusting the magnitude of the optical force. The real-time backscattering intensities of multiple cells are detected, and highly sensitive trapping is achieved. This microcone-shaped optical fiber probe is potentially a powerful device for dynamic cell assembly, optical sorting, and the precise diagnosis of vascular diseases.