The Lanczos method and its variants can be used to solve efficiently the rational interpolation problem. In this paper we present a suitable fast modification of a general look-ahed version of the Lanczos process in order to deal with polynomials expressed in the Chebyshev orthogonal basis. The proposed approach is particularly suited for rational interpolation at Chebyshev points, that is, at the zeros of Chebyshev polynomials. In fact, in this case it overcomes some of the numerical difficulties which limited the applicability of the look-ahed Lanczos process for determining the coefficients both of the numerators and of the denominators with respect to the standard power basis.