Despite significant progress made over the past two decades in the treatment of chronic myeloid leukemia (CML), there is still an unmet need for effective and safe agents to treat patients with resistance and intolerance to the drugs used in clinic. In this work, we designed 2-arylaminopyrimidine amides of isoxazole-3-carboxylic acid, assessed in silico their inhibitory potential against Bcr-Abl tyrosine kinase, and determined their antitumor activity in K562 (CML), HL-60 (acute promyelocytic leukemia), and HeLa (cervical cancer) cells. Based on the analysis of computational and experimental data, three compounds with the antitumor activity against K562 and HL-60 cells were identified. The lead compound efficiently suppressed the growth of these cells, as evidenced by the low IC 50 values of 2.8 ± 0.8 μM (K562) and 3.5 ± 0.2 μM (HL-60). The obtained compounds represent promising basic structures for the design of novel, effective, and safe anticancer drugs able to inhibit the catalytic activity of Bcr-Abl kinase by blocking the ATP-binding site of the enzyme.