Motion visually evoked potentials (mVEPs) have recently been explored as input features for brain-computer interfaces, in particular for the implementation of visual spellers. Due to low contrast and luminance requirements, motion-based intensification is less discomforting to the user than conventional approaches. So far, mVEP spellers were operated in the overt attention mode, wherein eye movements were allowed. However, the dependence on eye movements limits clinical applicability. Hence, the purpose of this study was to evaluate the suitability of mVEPs for gaze-independent communication. Sixteen healthy volunteers participated in an online study. We used a conventional speller layout wherein the possible selections are presented at different spatial locations both in the overt attention mode (fixation of the target) and the covert attention mode (central fixation). Additionally, we tested an alternative speller layout wherein all stimuli are sequentially presented at the same spatial location (foveal stimulation), i.e. eye movements are not required for selection. As can be expected, classification performance breaks down when switching from the overt to the covert operation. Despite reduced performance in the covert setting, conventional mVEP spellers are still potentially useful for users with severely impaired eye movements. In particular, they may offer advantages--such as less visual fatigue--over spellers using flashing stimuli. Importantly, the novel mVEP speller presented here recovers good performance in a gaze-independent setting by resorting to the foveal stimulation.