Cytoskeleton targeting agents are a group of chemotherapeutics used in the therapy of many types of cancer, such as breast, prostate, lung, bladder cancer, and others. At the same time, the assessment of the rheological properties of cancer cells is a relevant marker of their metastatic potential and therapeutic efficacy. For these reasons, understanding the interaction between the actin microfilament (MFs) network, microtubules (MTs), and so-called intermediate filaments (IFs) is crucial for the use of the rheological properties of cells as biomechanical markers. The current work compares the rheological properties of bladder cancer cells T24 and 5637, which differ in cytoskeletal composition, treated with a low dose of docetaxel (DTX) - a microtubule targeting agent (MTA). AFM revealed that 5637 cells stiffen over time when exposed to DTX, whereas changes in rheological properties of T24 cells are less pronounced, and both softening and stiffening of cells are observed. From immunostaining and Western blot analysis, we found that in addition to changes in the content and organization of MTs, reorganization of MFs and vimentin IFs also occurs. We show that both cell and nucleus morphology changes after DTX treatment. DTX treatment decreases and increases the migratory potential of 5637 and T24 cells, respectively. The current work shows that vimentin IFs modulate the nanomechanics of bladder cancer cells.