This paper proposes a bias reduction of the coefficients' estimator for linear regression models when observations are randomly censored and the error distribution is unknown. The proposed bias correction is applied to the weighted least squares estimator proposed by Stute [28] [W. Stute: Consistent estimation under random censorship when covariables are present. J. Multivariate Anal. \emph{45} (1993), 89-103.], and it is based on model-based bootstrap resampling techniques that also allow us to work with censored data. Our bias-corrected estimator proposal is evaluated and its behavior assessed in simulation studies concluding that both the bias and the mean square error are reduced with the new proposal.