Jump-diffusion models, such as Merton's and Kou's models, are essential in option pricing for capturing the discontinuous jump characteristics of asset prices. However, the nonlocal integral term in the resulting partial integro-differential equation (PIDE) presents substantial numerical challenges. This paper proposes an efficient and high-order numerical framework for pricing European and American options under these models. We employ the barycentric rational interpolation (BRI) collocation method for spatial discretization to achieve high-order accuracy and ensure numerical stability. For temporal discretization, the second-order implicit–explicit backward differentiation formula (IMEX-BDF2) is utilized, and the nonlocal integral term is evaluated via the Gauss–Legendre quadrature. To handle the early exercise constraint of American options, the operator splitting method is integrated into the framework. A rigorous error estimation and convergence analysis are provided to establish the mathematical reliability of the scheme. Numerical experiments demonstrate that the proposed method is highly efficient, providing accurate estimates for option values and stable numerical approximations for their Greeks (delta and gamma), validating its effectiveness for solving PIDEs in financial engineering.