The deformation and breakup of Newtonian and viscoelastic droplets are studied in confined shear flow. Our numerical approach is based on a combination of lattice-Boltzmann models and finite difference schemes, the former used to model two immiscible fluids with variable viscosity ratio and the latter used to model the polymer dynamics. The kinetics of the polymers is introduced using constitutive equations for viscoelastic fluids with finitely extensible nonlinear elastic dumbbells with Peterlin's closure. We quantify the droplet response by changing the polymer relaxation time τP, the maximum extensibility L of the polymers, and the degree of confinement, i.e., the ratio of the droplet diameter to wall separation. In unconfined shear flow, the effects of droplet viscoelasticity on the critical capillary number Cacr for breakup are moderate in all cases studied. However, in confined conditions a different behavior is observed: The critical capillary number of a viscoelastic droplet increases or decreases, depending on the maximum elongation of the polymers, the latter affecting the extensional viscosity of the polymeric solution. Force balance is monitored in the numerical simulations to validate the physical picture. © 2014 American Physical Society.