Optimization of sensor selection has been studied to monitor complex and large-scale systems with data-driven linear reduced-order modeling. An algorithm for greedy sensor selection is presented under the assumption of correlated noise in the sensor signals. A noise model is given using truncated modes in reduced-order modeling, and sensor positions that are optimal for generalized least squares estimation are selected. The determinant of the covariance matrix of the estimation error is minimized by efficient one-rank computations in both underdetermined and overdetermined problems. The present study also reveals that the objective function with correlated noise is neither submodular nor supermodular. Several numerical experiments are conducted using randomly generated data and real-world data. The results show the effectiveness of the selection algorithm in terms of accuracy in the estimation of the states of large-dimensional measurement data.