Elliptic interface problems with discontinuous coefficients appear in composite materials. The coefficient is usually highly oscillatory and may have abrupt jumps across the interface, leads to the pollution effect in the error. Compared with finite element or volume methods, finite difference methods (FDMs) avoid integrating high-frequency functions. Furthermore, the grid size requirement for high-order schemes is less stringent than low-order ones for the rapidly varying coefficient. We develop sixth-order FDMs for the elliptic interface problem with discontinuous variable coefficients on a rectangle. The FDMs utilize a 9-point compact stencil at any interior regular points and a 13-point stencil at irregular points near the interface Γ. For interior regular points away from Γ, we obtain sixth-order 9-point compact FDMs satisfying the M-matrix property for any mesh size h>0. We also derive sixth-order compact FDMs satisfying the M-matrix property for any h>0 under various Dirichlet/Neumann/Robin boundary conditions. For irregular points near Γ, we propose fifth-order 13-point FDMs, whose stencil coefficients can be effectively calculated by recursively solving several small linear systems.