In this paper we propose a novel method to construct confidence intervals in a class of linear inverse problems. First, point estimators are obtained via a spectral cut-off method depending on a regularisation parameter, that determines the bias of the estimator. Next, the proposed confidence interval corrects for this bias by explicitly estimating it based on a second regularisation parameter ρ, which is asymptotically smaller than α. The coverage error of the interval is shown to converge to zero. The proposed method is illustrated via two simulation studies, one in the context of functional linear regression, and the second one in the context of instrumental regression.