The neuronal circuitry that supports voluntary changes in eye position in tasks that require attention-driven oculo-motor control is well known. However, less is known about the neuronal basis for eye control during visual fixation. This, together with the fact that visual fixation is one of the most commonly used baseline conditions in resting-state functional magnetic resonance imaging (fMRI) studies, prompted us to conduct a study in which we employed resting-state fMRI and concurrent recordings of eye gaze to investigate the relationship between spontaneous changes in eye position during passive visual fixation and intrinsic brain activity. As a control experiment, we recorded fMRI brain activity related to cued voluntary vertical and horizontal changes in eye position in a block-related task-evoked fMRI experiment. Our results for the voluntarily performed changes in eye position elicited brain activity in the bilateral occipitotemporal cortex, supplementary motor cortex and frontal eye fields. In contrast, we show that slow fluctuations in eye position during passive visual fixation are linked to intrinsic brain activity, foremost in midline cortical brain regions located in the posteromedial parietal cortex and the medial prefrontal cortex, brain regions that act as core cortical hubs in the brain's default mode network. Our results suggest that subconscious and sustained changes in behavior are tied to intrinsic brain activity on a moment-by-moment basis.