Research over the past several decades has revealed that non-linguistic cognitive impairments can appear alongside language deficits in individuals with aphasia. One vulnerable cognitive domain is executive function, an umbrella term for the higher-level cognitive processes that allow us to direct our behavior towards a goal. Studies in healthy adults reveal that executive function abilities are supported by inner speech, the ability to use language silently in one's head. Therefore, inner speech may mediate the connection between language and executive function deficits in individuals with aphasia. Here, we investigated whether inner speech ability may link language and cognitive impairments in 59 adults with chronic, post-stroke aphasia. We used two approaches to measure inner speech: one based on internal retrieval of words and one based on internal retrieval plus silent manipulation of the retrieved phonological forms. Then, we examined relationships between these two approaches to measuring inner speech and five aspects of executive function ability: response inhibition, conflict monitoring/resolution, general task-switching ability, phonological control, and semantic control. We also looked for dissociations between inner speech ability and executive function ability. Our results show tentative relationships between inner speech (across multiple measurement approaches) and all aspects of executive function except for response inhibition. We also found evidence for a double dissociation: many participants show intact executive function despite poor inner speech, and vice versa, so neither process is strictly reliant on the other. We suggest that this work provides preliminary evidence of a bidirectional relationship between inner speech and executive function: inner speech supports some aspects of executive function via internal self-cueing and certain aspects of executive function support performance on complex inner speech tasks.