A CNT-modified viscoelastic locally resonant metamaterial (LRM) is investigated to study the complex dispersion curve, loss factor, wave transmission spectra, and effective stiffness. The dispersion curve analysis is based on the Bloch theorem applied to the nanocomposite LRM (NLRM) equations of motion and solving the resulting eigenvalue problem by the finite element method. The transmission spectra is obtained by COMSOL Multiphysics software. Due to the material viscoelasticity, multi-scale frequency-dependent homogenization of the nanocomposite is performed within the k(ω) framework which is systematically presented. The CNTs are added to the host LRM to enhance its stiffness without a significant increase in density. Hence, the impedance mismatch between the host nanocomposite and the coated cores is increased which affects the complex branch of NLRM, and shifts the imaginary part of the wave number to greater values without considerable change in its real part. So, the NLRM loss factor is increased which leads to lower wave transmission. Therefore, the NLRM exhibits simultaneously higher stiffness and loss factor with respect to ordinary LRMs. The effective stiffness of the CNT/host nanocomposite and the complex LRM dispersion curve and attenuation diagram are validated by experimental and numerical results, respectively. A parametric study is conducted to consider the effects of the CNT volume fraction (CNTVF), aspect ratio (AR), aggregation state and the CNT/host interface properties. Results show that the CNTVF and AR have the most influential effects on the NLRM properties. The CNT aggregation state is detrimental to the NLRM stiffness and wave attenuation properties too.