In this paper, we introduce the almost I-Hurewicz (AIH) property which is a simultaneous generalization of the I-Hurewicz (IH) (see [1, 5]) and the almost Hurewicz [23] properties. It is shown that the AIH property independent to the weakly I-Hurewicz (WIH) [6] property, where I is the proper admissible ideal of N. It is shown that in a regular space, the AIH property implies the IH property hence the WIH property, but not in Urysohn space. In a similar way, we consider almost Iγ-set and the almost star-I-Hurewicz property (ASIH) and it is shown that the image of almost Iγ-set under θ-continuous mapping has the AIH property