We already generalized the Rutishauser-Gragg-Harrod-Reichel algorithm for discrete least squares polynomial approximantion on the real axis to the rational case. In this paper, a new method for discrete least squares linearized rational approximation on the unit circle is presented. It generalizes the algorithm of Reichel-Ammar-Gragg for discrete least squares polynomial approximation on the unit circle to the rational case. The algorithm is fast in the sense that it requires order mα computation time where m is the number of data points and α is the degree of the approximant. We describe how this algorithm can be implemented in parallel. Examples illustrate the numerical behavior of the algorithm.