The disease atherosclerosis causes stenosis inside the patient's arteries, which often eventually turns lethal. Our goal is to detect a stenosis in a non-invasive manner, preferably in an early stage. To that end, we study whether and how laser speckle contrast imaging (LSCI) can be deployed. We start out by using computational fluid dynamics on a patient-specific stenosed carotid artery to reveal the flow profile in the region surrounding the stenosis, which compares well with particle image velocimetry experiments. We then use our own fully interferometric dynamic light scattering routines to simulate the process of LSCI of the carotid artery. Our approach offers an advantage over the established Monte Carlo techniques because they cannot incorporate dynamics. From the simulated speckle images, we extract a speckle contrast time series at different sites inside the artery, of which we then compute the frequency spectrum. We observe an increase in speckle boiling in sites where the flow profile is more complex, e.g., containing regions of backflow. In the region surrounding the stenosis, the measured speckle contrast is considerably lower due to the higher local velocity, and the frequency signature becomes notably different with prominent higher-order frequency modes that were absent in the other sites. Although future work is still required to make our new approach more quantitative and more applicable in practice, we have provided a first insight into how a stenosis might be detected in vivo using LSCI.