Combined saccade-vergence movements allow humans and other primates to align their eyes with objects of interest in three-dimensions. In the absence of saccades, vergence movements are typically slow, symmetrical movements of the two eyes in opposite directions. However, combined saccade-vergence movements produce vergence velocities that exceed values observed during vergence alone. This phenomenon is often called "vergence enhancement", or "saccade-facilitated vergence," though it is important to consider that rapid vergence changes, known as "vergence transients," are also observed during conjugate saccades. We developed a visual target array that allows monkeys to make saccades in all directions between targets spaced at distances that correspond to ~1° intervals of vergence angle relative to the monkey. We recorded the activity of vergence-sensitive neurons in the supra-oculomotor area (SOA), located dorsal and lateral to the oculomotor nucleus while monkeys made saccades with vergence amplitudes ranging from 0 to 10°. The primary focus of this study was to test the hypothesis that neurons in the SOA fire a high frequency burst of spikes during saccades that could generate the enhanced vergence. We found that individual neurons encode vergence velocity during both saccadic and non-saccadic vergence, yet firing rates were insufficient to produce the observed enhancement of vergence velocity. Our results are consistent with the hypothesis that slow vergence changes are encoded by the SOA while fast vergence movements require an additional contribution from the saccadic system. NEW & NOTEWORTHY Research into combined saccade-vergence movements has so far focused on exploring the saccadic neural circuitry, leading to diverging hypotheses regarding the role of the vergence system in this behavior. In this study, we report the first quantitative analysis of the discharge of individual neurons that encode vergence velocity in the monkey brain stem during combined saccade-vergence movements.