•Use metamodel to estimate the time-dependent failure probability (TDFP).•The proposed methods reduce computational cost drastically.•AK-co-IS is based on the optimal time-dependent design point.•AK-co-SS transforms small TDFP into a series of larger conditional ones. For efficiently estimating the time-dependent failure probability, two new methods named as the active learning Kriging (AK) coupled with importance sampling (AK-co-IS) and AK coupled with subset simulation (AK-co-SS) are proposed. The proposed methods are based on the fact that the AK coupled with Monte Carlo simulation (AK-MCS) method has been proved to be a very efficient method. However, for problem with small time-dependent failure probability or long service time, the size of candidate sample pool generated by MCS would be so large that the efficiency of AK-MCS is reduced. Therefore, the AK-co-IS and AK-co-SS are proposed to highly enhance the computational efficiency by greatly reducing the candidate sample pool size. And these two methods reduce the candidate sample pool size respectively by searching the optimal time-dependent design point to increase the ratio of failure samples and converting a rare event simulation problem into sequence of more frequent event ones. Through iteratively constructing the AK model to be convergent by the U-learning function in the IS and SS sample pools, respectively, the computational cost of estimating the time-dependent failure probability would reduce drastically compared with AK-MCS. Several examples are used to illustrate the efficiency and accuracy of the proposed methods.