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The study considers an iterative formulation of the ensemble Kalman filter (EnKF) for strongly nonlinear systems in the perfect-model framework. In the first part, a scheme is introduced that is similar to the ensemble randomized maximal likelihood (EnRML) filter by Gu and Oliver. The two new elements in the scheme are the use of the ensemble square root filter instead of the traditional (perturbed observations) EnKF and rescaling of the ensemble anomalies with the ensemble transform matrix from the previous iteration instead of estimating sensitivities between the ensemble observations and ensemble anomalies at the start of the assimilation cycle by linear regression. A simple modification turns the scheme into an ensemble formulation of the iterative extended Kalman filter. The two versions of the algorithm are referred to as the iterative EnKF (IEnKF) and the iterative extended Kalman filter (IEKF).