Bi-level thermal switches recently have gained much attention for control of heat conduction for thermal management applications. Here, we demonstrate a continuously tunable thermal switch using compressible graphene foams whose thermal conductance can be controlled by compression at different levels. A 1.2mm-thick graphene foam filtrated with 5% polymer is sandwiched in between two thermal contacts. The thermal conductivity of the graphene foam and the overall thermal resistance (including the graphene foam resistance and the two contact resistances) at different compression levels are measured by the miniaturized infrared reference bar method. The thermal conductivity increases from 0.056 W/m-K to 0.078 W/m-K, and the overall thermal resistance decreases from 0.0225 K-m2/W to 0.0056 K-m2/W, when the graphene foam is tuned from the uncompressed state to 85%-compressed state. The thermal switching ratio is 8.09. The experiments also show the promise of our continuously-variable thermal switch to regulate the hot side temperature as the cold size temperature and heat flux vary such as might be experienced by batteries in electric vehicles. The study promises novel and effective thermal management strategies for electronic devices with a continuously tunable, simple, and fast thermal switch.