Synchronous machines (SMs) represent nonlinear dynamical systems with stator flux linkages being crucial 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-the-art. However, DOB-FLE faces challenges in ensuring exponential convergence during transient conditions. 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 degree of freedom compared to the constant assumption imposed for DOB-FLE and allowing for the estimation of both the extended states and the flux linkages through an ESO. Simulation results from a 35-kW SM drive demonstrate that ESO-FLE achieves exponential performance under transient conditions compared to DOB-FLE.