In some situations, the failure time of interest is defined as the gap time between two related events and the observations on both event times can suffer either right or interval censoring. Such data are usually referred to as doubly censored data and frequently encountered in many clinical and observational studies. Additionally, there may also exist a cured subgroup in the whole population, which means that not every individual under study will experience the failure time of interest even-tually. In this paper, we consider regression analysis of doubly censored data with a cured subgroup under a wide class of flexible transformation cure models. Specifically, we consider marginal likeli-hood estimation and develop a two-step approach by combining the multiple imputation and a new expectation-maximization (EM) algorithm for its implementation. The resulting estimators are shown to be consistent and asymptotically normal. The finite sample performance of the proposed method is investigated through simulation studies. The proposed method is also applied to a real dataset arising from an AIDS cohort study for illustration.