The superconducting proximity effect in graphene can be used to create Josephson junctions with critical currents that can be tuned using local field-effect gates. These junctions have the potential to add functionality to existing technologies; for example, superconducting quantum interference device (SQUID) magnetometers with adaptive dynamic range and superconducting qubits with fast electrical control. Here, we present measurements of graphene-based superconducting quantum interference devices incorporating ballistic Josephson junctions that can be controlled individually. We investigate the magnetic field response of the SQUIDs as the junctions are gated and as the device is tuned between symmetric and asymmetric configurations. We find a highest transfer function ≈ 300 μV/Φ0, which compares favorably with conventional, low temperature DC SQUIDs. With low noise readout electronics and optimised geometries, devices based on ballistic graphene Josephson junctions have the potential to match the sensitivity of traditional SQUIDs while also providing additional functionality.