It remains unknown whether attention selects information 'early' or 'late' in the information-processing stream. Object-based attention (OBA) is well-positioned to shed light on this long-standing debate given that some studies have reported OBA effects even when object percepts are incomplete, suggesting that perceptual object formation happens prior to OBA deployment (though, most data suggest a later selection). Here, we aimed to ascertain the level within the visual cortical hierarchy (V1-V4, etc.) at which effects of OBA are first observed. While acquiring fMRI data, we asked participants to identify a target that was preceded by a central arrow cue (60% valid) in the double-rectangle paradigm (cf. Egly et al, 1994). The rectangles were either occluded or visible (un-occluded). Behaviorally, participants identified the target faster when it appeared on the cued object versus the uncued object. This same-object advantage emerged irrespective of object occlusion. We independently localized retinotopically-specific regions of cortex corresponding to target locations (object ends) to examine neural fluctuations in each region of the visual cortical hierarchy (both dorsal and ventral aspects). Consistent with our behavioral data, following target onset, activation in early visual areas increased at locations on the cued object, but not the uncued object; indicative of a neural same-object advantage in these regions. Specifically, we found that cue-evoked OBA effects are not evident in V1, emerging in V2 and V3, whereas target-evoked OBA effects are strongest in V1, but weaken in V2 and V3. Our results also indicate that cue-evoked object-based effects differ between the visible and occluded conditions, suggesting that object selection may vary along the visual cortical hierarchy. Thus, the level of the visual cortical hierarchy at which effects of OBA are observed may be flexible (early vs. late), depending on whether the target of selection (here, an object) comprises a completed percept.