We adopt the least squares Monte Carlo (LSMC) method to price time-capped American options. The cap can be an independent random variable or dependent on the asset price at a random time. We investigate various time caps. In particular, we give an algorithm for pricing the American options capped by the first drawdown epoch, focusing on the geometric Lévy market. We prove that our estimator converges to the true price as the discretization step tends to zero and the number of trajectories tends to infinity.