The authors extend some interesting results proved earlier by themselves in the univariate case to the bivariate one. They also remark that the results can easily be extended for \(m\) \((> 2)\) variables, too. More precisely they prove that the bivariate interpolation polynomials of Hermite-Fejér based on the Chebyshev nodes of the first kind, those of Lagrange based on the Chebyshev nodes of second kind and \(\pm 1\), and those of bivariate Shepard operators, have the property of partial preservation of global smoothness, with respect to various bivariate moduli of continuity.