Synchronous machines (SMs) are characterized as nonlinear dynamic systems, with stator flux linkages being critical for controller design. Among the developed methods for flux linkage estimation, the disturbance observer-based flux linkage estimator (DOB-FLE) is recognized as the state-of-theart. However, DOB-FLE faces challenges in ensuring exponential convergence during transient states. To address this, this paper introduces an extended state observer-based flux linkage estimator (ESO-FLE), which represents an advancement over DOB-FLE. This novel approach utilizes extended states to model the nonlinear disturbance term as time-varying ramp signals, offering a contrast to the constant assumption employed in DOB-FLE. It concurrently estimates both the extended states and the flux linkages through an ESO. Simulation results from a 35-kW SM drive demonstrate that ESO-FLE provides superior transient performance under dynamic conditions compared to DOB-FLE.