Methods for the evaluation of the predictive accuracy of biomarkers with respect to survival outcomes subject to right censoring have been discussed extensively in the literature. In cancer and other diseases, survival outcomes are commonly subject to interval censoring by design or due to the follow up schema. In this article, we present an estimator for the area under the time-dependent receiver operating characteristic (ROC) curve for interval censored data based on a nonparametric sieve maximum likelihood approach. We establish the asymptotic properties of the proposed estimator and illustrate its finite-sample properties using a simulation study. The application of our method is illustrated using data from a cancer clinical study. An open-source R package to implement the proposed method is available on Comprehensive R Archive Network.