This paper proposes the finite difference-barycentric interpolation collocation scheme for the nonlinearly coupled Burgers equation. For the coupled Burgers equation, it is firstly transformed into a linear equations via the direct linearization iteration. The spatial direction is discretized by the barycentric interpolation collocation method, and the time direction is discretized by the Crank-Nicolson scheme. The corresponding linear algebraic equations are derived. The consistency analysis is provided for the semi-discrete scheme in space and the fully discrete scheme. Numerical experiments verify the high precision and efficiency of our scheme.