We introduce the finite bivariate biorthogonal M-Jacobi polynomials with the help of the finite orthogonal M polynomials and the biorthogonal polynomials suggested by the Jacobi polynomials. Then, by adding two further parameters the generalized finite bivariate biorthogonal M-Jacobi polynomials are defined, and a set of finite bivariate biorthogonal functions is obtained using this generalized polynomials. On the other hand, we investigate some relationships between these polynomials and recently defined families and give some recurrence relations.