Manipulating cells at a small scale is widely acknowledged as a complex and challenging task, especially when it comes to cell grasping and transportation. Various precise methods have been developed to remotely control the movement of microrobots. However, the manipulation of micro-objects necessitates the use of end-effectors. This paper presents a study on the control of movement and grasping operations of a magnetic microrobot, utilizing only 3 pairs of electromagnetic coils. A specially designed microgripper is employed on the microrobot for efficient cell grasping and transportation. To ensure precise grasping, a bending deformation model of the microgripper is formulated and subsequently validated. To achieve precise and reliable transportation of cells to specific positions, an approach that combines an extended Kalman filter with a model predictive control method is adopted to accomplish the trajectory tracking task. Through experiments, we observe that by applying the proposed control strategy, the mean absolute error of path tracking is found to be less than 0.155 mm. Remarkably, this value accounts for only 1.55% of the microrobot’s size, demonstrating the efficacy and accuracy of our control strategy. Furthermore, an experiment involving the grasping and transportation of a zebrafish embryonic cell (diameter: 800 μm) is successfully conducted. The results of this experiment not only validate the precision and effectiveness of the proposed microrobot and its associated models but also highlight its tremendous potential for cell manipulation in vitro and in vivo.