A tunable near-infrared surface plasmon resonance (SPR) biosensor based on gate-controlled graphene plasmons is numerically investigated by using the finite element method (FEM) and the transfer matrix method (TMM). The novel properties 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 resonance wavelengths of 1100, 1200, 1310, 1550, 1700 and 1900 nm. The figure of merit (FOM) achieves 129.3, 101.1, 84.5, 67.7, 69.5 and 59.7 eV−1, respectively, at these resonance wavelengths. 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 wavelengths of 1100, 1200, 1310, 1550, 1700 and 1900 nm. The FOM achieves 135.6, 103.0, 88.9, 62.2, 66.6 and 61.5 eV−1, respectively, at these incident wavelengths. Theoretical estimates suggest that the limit of detection (LOD) of the sensor’s DNA sensing can reach the level of femtomolar or even attomolar, 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.