Awareness of action is a pervasive personal experience that is crucial in understanding self-generated and other-generated actions as well as their effects. A large body of research suggests that action awareness, as measured by the magnitude of temporal binding between an action and its effect in an operant action task (i.e., intentional binding), is rooted in the human capacity to experience self-agency and establish action intentions. Whereas previous research mainly addressed the role of intentionality itself in these socially well-shared experiences, in the present study we focused specifically on one important aspect of it: the quality or strength of action intentions. We expected and established that stronger intentions increase intentional binding. Specifically, the magnitude of the binding effect, as assessed by the Libet clock task in which two actions were followed by the same neutral tone, was elevated for the action that was enacted with stronger intentions. We briefly discuss the implications of the observed role of intentional strength in temporal binding between action and effect, for promoting a better understanding and examination of how the concept of intentionality is associated with action awareness in general, and the experience of being the agent of one's own actions in particular.