Biophysical models of diffusion in gray matter (GM) can provide unique information about microstructure of the human brain, in health and disease. Therefore, their compatibility with clinical settings is key. Neurite Exchange Imaging (NEXI) is a two-compartment model of GM microstructure that accounts for inter-compartment exchange, whose parameter estimation requires multi-shell multi-diffusion time data. In this work, we report the first estimates of NEXI in human cortex obtained on a clinical MRI scanner. To do that, we establish an acquisition protocol and fitting routine compatible with clinical scanners. The model signal equation can be expressed either in the narrow-pulse approximation, NEXINPA, or accounting for the actual width of the diffusion gradient pulses, NEXIWP. While NEXINPA enables a faster analytical fit and is a valid approximation for data acquired on high-performance gradient systems (preclinical and Connectom scanners), on which NEXI was first implemented, NEXIWP has significant relevance for data acquired on clinical scanners with longer gradient pulses. We establish that, in the context of broad pulses, NEXIWP estimates were more comparable to previous literature values. Furthermore, we evaluate the repeatability of NEXI estimates in the human cortex on a clinical MRI scanner and show intra-subject variability to be lower than inter-subject variability, which is promising for characterizing healthy and patient cohorts. Finally, we analyze the relationship of NEXI parameters on the cortical surface to the Myelin Water Fraction (MWF), estimated using an established multicomponent T2 relaxation technique. Indeed, although it is present in small quantities in the cortex, myelin can be expected to decrease permeability. We confirm a strong correlation between the exchange time (tex) estimates and the MWF, although the spatial correspondence between the two is brain-region specific and other drivers of tex than myelin density are likely at play.