The cytoskeleton comprises microtubules, actin microfilaments, and intermediate filaments. The cytoskeleton plays a role in the mechanical function of cell adhesion, motility, and migration. Currently, cancer cell mechanics is an important topic of understanding cancer progression. Atomic force microscopy (AFM) is one of the techniques used to determine the mechanical properties of cells. In this study, AFM was used to measure the force curves of U937 cancer cells treated with two types of anticancer drugs (colchicine and taxol), and alterations in the Young’s modulus were subsequently investigated. The study results indicate that the force curve trend was dissimilar among the three types of cells (cells in the control group and in two types of anticancer drugs). In addition, when using the AFM to analyze the influence of drugs on the mechanical properties of cells, applying a small loading force from the AFM tip (small cell indentation depth) produced more accurate results. At a small cell indentation depth, the Young’s modulus order was Ecolchicine < Econtrol < Etaxol because colchicine caused the cell microtubules to disassemble, thereby reducing cell strength, whereas taxol caused the microtubules to assemble, which increased cell strength. At a large cell indentation depth, the Ecolchicine value was the largest because colchicine reduced the strength and height of the cells and the effect of the underlying substrate was reflected in and influenced the force curve.