The study of drop deformation in response to various stresses has long piqued the interest of several academics. The deformation behavior of cells, drug carriers, and even drug particles moving via microcapillaries inside the human body can be modeled using a viscoelastic drop model. A drop breakup study can also provide better design guidance for nanocarriers that can deliver on-demand burst drug releases at specific cancer sites. Thus, we attempted to investigate the deformation and breakup of a shear-thinning finitely extensible nonlinear elastic-peterlin (FENE-P) drop moving through the constricted microchannel. The computational simulation suggested that drop deformation and breakup can be manipulated by varying of parameters like channel confinement, Deborah number, solvent viscosity ratio, viscosity ratio, and capillary number. We attempted to find the critical capillary number for initiation of drop breakup. Observations from present study will give valuable insights into deformation and breakup patterns of drug carriers inside constricted microcapillaries. The simulations of the two-phase viscoelastic drop─Newtonian matrix system were performed on an open-source solver, Basilisk.