Rational Bernstein operators are widely used in approximation theory and geometric modeling but in general they do not reproduce linear polynomials. Based on the work of P. Piţul and P. Sablonnière, we construct a new family of triangular and tensor product bivariate rational Bernstein operators, which are positive and reproduce the linear polynomials. The main result is a proof of convergence of the bivariate rational Bernstein operators defined on the square or triangle.