Near-field radiative heat transfer (NFRHT) is strongly related with many applications such as near-field imaging, thermo-photovoltaics and thermal circuit devices. The active control of NFRHT is of great interest since it provides a degree of tunability by external means. In this work, a magnetically tunable multiband NFRHT is revealed in a system of two suspended graphene sheets at room temperature. It is found that the single-band spectra for B=0 split into multiband spectra under an external magnetic field. Dual-band spectra can be realized for a modest magnetic field (e.g., B=4T). One band is determined by intraband transitions in the classical regime, which undergoes a blue shift as the chemical potential increases. Meanwhile, the other band is contributed by inter-Landau-level transitions in the quantum regime, which is robust against the change of chemical potentials. For a strong magnetic field (e.g., B=15T), there is an additional band with the resonant peak appearing at near-zero frequency (microwave regime), stemming from the magnetoplasmon zero modes. The great enhancement of NFRHT at such low frequency has been little reported. This work may pave a way for multiband thermal information transfer based on atomically thin graphene sheets.