In daily life, activities requiring the hand and eye to work separately are as frequent as activities requiring tight eye-hand coordination, and we effortlessly switch from one type of activity to the other. Such flexibility is unlikely to be achieved without each effector "knowing" where the other one is at all times, even when it is static. Here, we provide behavioral evidence that the mere position of the static hand affects one eye movement parameter: saccadic reaction time. Two monkeys were trained and 11 humans instructed to perform non-delayed or delayed visually-guided saccades to either a right or a left target while holding their hand at a location either near or far from the eye target. From trial to trial, target locations and hand positions varied pseudorandomly. Subjects were tested both when they could and when they could not see their hand. The main findings are: (1) the presence of the static hand in the workspace did affect saccade initiation, (2) this interaction persisted when the hand was invisible, (3) it was strongly influenced by the delay duration: hand-target proximity retarded immediate saccades, while it could hasten delayed saccades and, (4) this held true for humans, as well as for each of the two monkeys. We propose that both visual and non visual hand position signals are used by the primates' oculomotor system for the planning and execution of saccades, and that this may result in an hand-eye competition for spatial attentional resources that explains the delay-dependent reversal observed.