Verbal short-term memory (STM) deficits are associated with language processing impairments in people with aphasia. Importantly, the integrity of STM can predict word learning ability and anomia therapy gains in aphasia. While the recruitment of perilesional and contralesional homologous brain regions has been proposed as a possible mechanism for aphasia recovery, little is known about the white-matter pathways that support verbal STM in post-stroke aphasia. Here, we investigated the relationships between the language-related white matter tracts and verbal STM ability in aphasia. Nineteen participants with post-stroke chronic aphasia completed a subset of verbal STM subtests of the TALSA battery including nonword repetition (phonological STM), pointing span (lexical-semantic STM without language output) and repetition span tasks (lexical-semantic STM with language output). Using a manual deterministic tractography approach, we investigated the micro- and macrostructural properties of the structural language network. Finally, we assessed the relationships between individually extracted tract values and verbal STM scores. We found significant correlations between volume measures of the right Uncinate Fasciculus and all three verbal STM scores. We also found significant associations between verbal STM scores and the left long segment of the Arcuate, the left Uncinate and the right Inferior Fronto-Occipital Fasciculi, although these did not survive FDR corrections. These findings suggest that the integrity of the right UF is associated with phonological and lexical-semantic verbal STM ability in aphasia and highlight the potential compensatory role of right-sided ventral white matter language tracts in supporting verbal STM after aphasia-inducing left hemisphere insult.