The assimilation of high-quality in situ data into ocean models is known to lead to imbalanced analyses and spurious circulations when the model dynamics or the forcing contain systematic errors. Use of a bias estimation and correction scheme has been shown to significantly improve the balance of the analysed states in such cases. Given the large impact of zonal wind stress bias on the tropical ocean circulation we propose to estimate directly a bias in the forcing rather than in the model state vector. The bias scheme introduced by Dee and Da Silva is modified for use with the Ensemble Kalman Filter and approximate schemes are derived to achieve computational savings. Twin experiments show that for this particular application these simplified schemes efficiently reproduce the bias. The bias in the wind stress can be completely reconstructed sequentially, also when it is slowly evolving in time, leading to well-balanced analyses and unbiased forecasts. In particular the correct recovery of the amplitude of the wind stress bias is found to be important. We additionally investigate the use and correction of a mean sea level climatology in the context of forcing bias estimation and suggest an alternative approach based on the ensemble mean sea level.