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In this article, the enumeration method and the expectation maximization (EM) algorithm are combined for the identification of linear parameter varying (LPV) systems with unknown time-varying system delays. The unknown system delays are addressed as the latent variables which are assumed to be uniformly distributed in the possible value range. At each sampling instant, by using the basic idea of the enumeration method, the unknown delay is estimated by maximizing its posterior distribution function (PDF). Moreover, the problem of outliers is also considered in the identification procedure and the robust t-distribution instead of the common Gaussian distribution is used to handle the data abnormality. All the equations to estimate the unknown parameters and the delays are all derived by using the EM algorithm. The validity and robustness of the proposed approach are demonstrated by using several simulation examples.