The main purpose of the paper is to show how a magnetoresistive (MR) element can work as a current sensor instead of using a Wheatstone bridge composed by four MR elements, defining the concept of a magnetoresistive shunt (MR-shunt). This concept is reached by considering that once the MR element is biased at a constant current, the voltage drop between its terminals offers information, by the MR effect, of the current to be measured, as happens in a conventional shunt resistor. However, an MR-shunt has the advantage of being a non-dissipative shunt since the current of interest does not circulate through the material, preventing its self-heating. Moreover, it provides galvanic isolation. First, we propose an electronic circuitry enabling the utilization of the available MR sensors integrated into a Wheatstone bridge as sensing elements (MR-shunt). This circuitry allows independent characterization of each of the four elements of the bridge. An independently implemented MR element is also analyzed. Secondly, we propose an electronic conditioning circuit for the MR-shunt, which allows both the bridge-integrated element and the single element to function as current sensors in a similar way to the sensing bridge. Third, the thermal variation in the sensitivity of the MR-shunt, and its temperature coefficient, are obtained. An electronic interface is proposed and analyzed for thermal drift compensation of the MR-shunt current sensitivity. With this hardware compensation, temperature coefficients are experimentally reduced from 0.348%/°C without compensation to -0.008%/°C with compensation for an element integrated in a sensor bridge and from 0.474%/°C to -0.0007%/°C for the single element.