White matter (WM) abnormalities are commonly reported in schizophrenia but whether these arise from the axon or myelin compartments or both is not known. In addition, the relationship between WM abnormalities and cognitive function is not fully explored in this condition. We recruited 39 individuals with schizophrenia spectrum disorders and 37 healthy comparison subjects. All participants underwent MRI scanning at 4 Tesla to collect data in the prefrontal white matter on magnetization transfer ratio (MTR) and diffusion tensor spectroscopy (DTS) which provide information on myelin and axon compartments, respectively. We also collected Matrics Composite Cognitive Battery (MCCB) and Stroop cognitive data. We found an elevated N-acetylaspartate (NAA) apparent diffusion coefficient in schizophrenia in this cohort as in our previous work; we also observed poorer performance on both the MCCB composite and the Stroop in schizophrenia patients compared to controls. The MTR measure was correlated with the MCCB composite (r = 0.363, p = 0.032) and Stroop scores (r = 0.387, p = 0.029) in healthy individuals but not in schizophrenia. Since this is the first exploration of the relationship between these WM and cognitive measures, we consider our analyses exploratory and did not adjust for multiple comparisons; the findings are not statistically significant if adjusted for multiple comparisons. These findings indicate that WM integrity is associated with cognitive function in healthy individuals but this relationship breaks down in patients with schizophrenia.