A tunable near-infrared surface plasmon resonance (SPR) biosensor based on gate-controlled graphene plasmons is investigated theoretically. The novel characteristics of chemical potential sensing make the proposed sensor promising in the application of ultra-sensitive and highly specific biosensing technology. The sensitivity of chemical potential sensing in wavelength interrogation mode can be calculated to be 1.5, 1.89, 2.29, 3.21, 3.73 and 4.68 nm/meV respectively at the central wavelength of 1100, 1200, 1310, 1550, 1700 and 1900 nm. The much smaller full width half maximum (FWHMs) comparing with that of 2D nanomaterial-enhanced metal SPR sensors indicates higher figure of merit. The sensitivity of chemical potential sensing in gate voltage interrogation mode also can be calculated to be 156.9822, 143.6147, 131.0779, 111.0351, 101.3415 and 90.6038 mV/meV respectively at the incident wavelength of 1100, 1200, 1310, 1550, 1700 and 1900 nm. It can be estimated theoretically that the limit of detection (LOD) in DNA sensing of the proposed sensor can reach femtomolar level and even attomolar level, comparable to and even lower than that of 2D nanomaterial-enhanced metal SPR sensors with AuNPs as a sensitivity enhancement strategy. The feasibility of preparation and operation of this new concept SPR biosensor is also analyzed and discussed.