Although synchrony between the limbs is an often-cited feature of bimanual coordination, recent studies have also highlighted the small asynchronies that can occur. The visuo-motor demands of any bimanual task are considered central to the emergence of asynchrony, but the relationship between the two remains largely unexplored. This study aimed to address this issue. Hand and eye movements were measured in 19 participants, while they made either unimanual or bimanual reach-to-point (aiming) movements to targets presented on a touchscreen. Bimanual movements were either congruent (same-sized targets) or incongruent (different-sized targets). Resulting hand data showed many of the typical patterns of movement previously reported. While temporal coupling between the limbs remained largely evident for bimanual movements, small between-limb asynchronies were apparent and demonstrated clear associations with the competing precision requirements of the targets and related visual attention. Participants mainly directed their gaze towards the more difficult target with corresponding reaching movements demonstrating greater precision than for the easier target. Additionally, there was a reliable tendency for the hand reaching towards the more difficult target to lead. Importantly, it was the competing visuo-motor demands of individual movements rather than overall difficulty that resulted in greater between-limb asynchrony; accordingly, where both targets were small (i.e., the most difficult condition), asynchrony was significantly less pronounced than for incongruent bimanual conditions. The results show how the visuo-motor system balances its apparent drive for synchrony in coordinating bimanual movements with the competing demands that characterise the constituent unimanual movements.