Recent studies by Bastiaanse and colleagues found that time reference is selectively impaired in people with nonfluent agrammatic aphasia, with reference to the past being more difficult to process than reference to the present or to the future. To account for this dissociation, they formulated the PAst DIscourse LInking Hypothesis (PADILIH), which posits that past reference is more demanding than present/future reference because it involves discourse linking. There is some evidence that this hypothesis can be applied to people with fluent aphasia as well. However, the existing evidence for the PADILIH is contradictory, and most of it has been provided by employing a test that predominantly taps retrieval processes, leaving largely unexplored the underlying ability to encode time reference-related prephonological features. Within a cross-linguistic approach, this study tests the PADILIH by means of a sentence completion task that 'equally' taps encoding and retrieval abilities. This study also investigates if the PADILIH's scope can be extended to fluent aphasia. Greek- and Italian-speaking individuals with aphasia participated in the study. The Greek group consisted of both individuals with nonfluent agrammatic aphasia and individuals with fluent aphasia, who also presented signs of agrammatism. The Italian group consisted of individuals with agrammatic nonfluent aphasia only. The two Greek subgroups performed similarly. Neither language group of participants with aphasia exhibited a pattern of performance consistent with the predictions of the PADILIH. However, a double dissociation observed within the Greek group suggests a hypothesis that may reconcile the present results with the PADILIH.