Human learners can resolve referential ambiguity and discover the relationships between words and meanings through a cross-situational learning (CSL) strategy. Some people with aphasia (PWA) can learn word-referent pairings under referential uncertainty supported by online feedback. However, it remains unknown whether PWA can learn new words cross-situationally and if such learning ability is supported by statistical learning (SL) mechanisms. The present study examined whether PWA can learn novel word-referent mappings in a CSL task without feedback. We also studied whether CSL is related to SL in PWA and neurologically healthy individuals. We further examined whether aphasia severity, phonological processing and verbal short-term memory (STM) predict CSL in aphasia, and also whether individual differences in verbal STM modulate CSL in healthy older adults. Sixteen people with chronic aphasia underwent a CSL task that involved exposure to a series of individually ambiguous learning trials and a SL task that taps speech segmentation. Their learning ability was compared to 18 older controls and 39 young adults recruited for task validation. CSL in the aphasia group was below the older controls and young adults and took place at a slower rate. Importantly, we found a strong association between SL and CSL performance in all three groups. CSL was modulated by aphasia severity in the aphasia group, and by verbal STM capacity in the older controls. Our findings indicate that some PWA can preserve the ability to learn new word-referent associations cross-situationally. We suggest that both PWA and neurologically intact individuals may rely on SL mechanisms to achieve CSL and that verbal STM also influences CSL. These findings contribute to the ongoing debate on the cognitive mechanisms underlying this learning ability.