An inerter is a mechanical analogue to a capacitor, where the force across the device is proportional to relative, rather than absolute, acceleration. This concept can offer attractive performance in a wide variety of engineering vibration problems, because the engineer can tune the device without dramatically increasing the physical mass of the structure. Consequently, there have been many studies over the last two decades that have explored their application to bridge vibrations, seismic isolation of tall buildings, vehicle suspensions, and other engineering problems. Several configurations of inerter systems have been proposed, typically involving the inerter in a vibration absorber, or by using the inerter as part of an isolation system. However, to date there have been limited studies that have explored the combination of inerters with semi-active devices such as magnetorheological fluid dampers. Furthermore, because one manifestation of inerters involves the use of hydraulic fluid, it is possible for magnetorheological effects to be integrated into the inerter itself. The present study investigates the feasibility of this approach for practical scenarios. A quasi-static model is developed, combining an existing model of a fluid inerter with simplified models for magnetorheological fluids. The trade-off between damping performance and inerter performance is explored. The model is then used in a case study, where its potential use in a control strategy known as a parallel-layout inerter damper is investigated.