The finite-element method was employed to analyze axisymmetric creeping motion of a deformable drop in the entrance region of a cylindrical tube. In the present study a non-Newtonian drop suspended in a non-Newtonian medium was considered, for which a truncated power-law model was employed. The penalty function method was used to eliminate the pressure variables from the system equations, considerably reducing the computational time required. In solving the system equations an unstructured mesh generator and auto-remeshing technique were used to describe the shapes of highly deformed drops. The results are in good agreement with experiments of Chin and Han [J. Rheol. 23, 557-590 (1979); 24, 1-37 (1980)] describing the deformation as the drop approaches the inlet from the reservoir section and recoil after it enters the tube. The present study demonstrates the effects of the capillary number and viscosity ratio on the deformation of non-Newtonian drops suspended in non-Newtonian liquids in the entrance region of a cylindrical tube.