In the present study, the linear and nonlinear free vibration behaviors of the carbon nanotubes (CNTs)-reinforced multiphase magneto-electro-elastic (MEE) deep plane-curved beams are investigated. The MEE material is composed of Barium Titanate as a piezoelectric and Cobalt ferrite as a piezomagnetic. The uniform and different functionally graded (FG) distributions of the CNTs along the beam thickness direction are considered. The rule of mixture is used to estimate the effective material properties of the CNT-reinforced MEE (CNT-MEE) beams. The governing equations are derived based on the first-order shear deformation beam theory (FSDBT) along with von Kármán’s geometric nonlinearity assumptions by applying Hamilton’s principle. The space domain is discretized using the differential quadrature method (DQM). Thereafter, harmonic balance and direct-iterative methods are used to obtain the linear and nonlinear natural frequencies of the beam. To validate the present approach, its fast convergence rate is verified and the numerical results for the different straight and curved beams are compared with those available in the literature. Finally, the influences of various parameters such as electrical and magnetic fields, piezoelectric volume fraction, volume fraction and the distribution patterns of the CNTs, opening angle of the beam, ratio of length-to-radius of gyration, nonlinear vibration amplitudes, and different boundary conditions on the linear and nonlinear fundamental frequencies of the CNT-MEE deep plane-curved beams are studied. It is found that by considering the magnetic effects, in some cases, the linear and nonlinear fundamental frequencies increase about 3.1% and 7.2%, respectively.