The mechanism of action for the cerebral vasculature hemodynamic response to the electric field effects during transcranial direct current stimulation (tDCS) has been less explored. We postulate that such a mechanistic understanding of the cerebrovascular reactivity (CVR) to tDCS can facilitate adequate tDCS dosing to facilitate cognitive rehabilitation in mild cognitive impairment and dementia as well as motor rehabilitation in stroke and traumatic brain injuries. This study presents a system identification approach to evaluate CVR under high-definition (HD) tDCS using a physiologically constrained model based on a multi-compartmental neurovascular system of vascular smooth muscle, perivascular space, synaptic space, and astrocyte glial cell. The physiologically detailed model generated vessel oscillations in the frequency range of 0.05 – 0.2 Hz driven by nonlinear calcium dynamics. Then, model linearization was performed to develop a grey-box linear model for evaluating the acute effects of HD-tDCS based on the CVR data from healthy humans. CVR was measured using functional near-infrared spectroscopy (fNIRS) with optodes in the vicinity of 4x1 HD-tDCS electrodes. The grey-box linear model response of vessel response through the synaptic potassium pathway was found comparable to known hemodynamic responses. Then, the grey-box linear model was fitted to the CVR to anodal HD-tDCS found from the normalized changes in the fNIRS-total haemoglobin (tHb) concentration (blood volume) in eleven healthy participants. We found that the model pathway from perivascular potassium to the vessel circumference presented the best fNIRS-tHb fit with the least mean square error. The fitted grey-box linear model response presented an initial dip in the vessel response that provided a mechanistic understanding of the fNIRS-tHb changes at the onset of HD-tDCS, which needs to be validated in the future with functional magnetic resonance imaging in conjunction with HD-tDCS.