In the present study, a multi-pseudo-potential model is used to simulate the deformation and breakup of bubbles and droplets under simple shear flow. It is shown that the current model can adjust the amount of surface tension, independent of the interface thickness, equation of state (EOS), and reduced temperature. Considering the available findings, no comprehensive study has been performed on all aspects of deformation of bubbles and droplets under shear flow using numerical methods. Bubble or droplet deformation under simple shear flow depends on two non-dimensional numbers: capillary number (Ca) and viscosity ratio $(λ).$ In this investigation, various scenarios, including small deformation, large deformation, and breakup for bubbles $(0.2 < λ < 1)$ and droplets $(1 < λ < 5),$ are separately studied. Results of the multi-pseudo-potential model show that the bubble and droplet deformations oscillate under shear flow and undergo elongation and contraction over time to converge to the final shape. As the capillary increases by more than one, the bubble expands and shrinks. The bubble tips become sharp, and so-called slender, at a low viscosity ratio $(λ).$ A detailed comparison is made between the numerical results for deformation parameters of the present model and the experimental results available in the references.