When reaching for an object with the hand, the gaze is usually directed at the target. In a laboratory setting, fixation is strongly maintained at the reach target until the reaching is completed, a phenomenon known as "gaze-anchoring". While conventional accounts of such tight eye-hand coordination have often emphasized the internal synergetic linkage between both motor systems, more recent optimal control theories regard motor coordination as the adaptive solution to task requirements. We here investigated to what degree gaze control during reaching is modulated by task demands. We adopted a gaze-anchoring paradigm in which participants had to reach for a target location. During the reach, they additionally had to make a saccadic eye movement to a salient visual cue presented at locations other than the target. We manipulated the task demands by independently changing reward contingencies for saccade reaction time (RT) and reaching accuracy. On average, both saccade RTs and reach error varied systematically with reward condition, with reach accuracy improving when the saccade was delayed. The distribution of the saccade RTs showed two types of eye movements: fast saccades with short RTs, and voluntary saccade with longer RTs. Increased reward for high reach accuracy reduced the probability of reflexive fast saccades, but left their latency unchanged. The results suggest that gaze-anchoring acts through a suppression of fast saccades, a mechanism that can be adaptively adjusted to the current task demands.