Global neuronal workspace theory predicts that damage to long-distance white matter (WM) tracts should impair access to consciousness during the perception of brief stimuli. To address this issue, we studied visual backward masking in 18 patients at the very first clinical stage of multiple sclerosis (MS), a neurological disease characterized by extensive WM damage, and in 18 matched healthy subjects. In our masking paradigm, the visibility of a digit stimulus increases non-linearly as a function of the interval duration between this target and a subsequent mask. In order to characterize quantitatively, for each subject, the transition between non-conscious and conscious perception of the stimulus, we used non-linear regression to fit a sigmoid curve to objective performance and subjective visibility reports as a function of target-mask delay. The delay corresponding to the inflexion point of the sigmoid, where visibility suddenly increases, was termed the "non-linear transition threshold" and used as a summary measure of masking efficiency. Objective and subjective non-linear transition thresholds were highly correlated across subjects in both groups, and were higher in patients compared to controls. In patients, variations in the non-linear transition threshold were inversely correlated to the Magnetization transfer ratio (MTR) values inside the right dorsolateral prefrontal WM, the right occipito-frontal fasciculus and the left cerebellum. This study provides clinical evidence of a relationship between impairments of conscious access and integrity of large WM bundles, particularly involving prefrontal cortex, as predicted by global neuronal workspace theory.