Nanocomposites and magnetic field effects on nanostructures have received great attention in recent years. A large amount of research work was focused on developing the proper theoretical framework for describing many physical effects appearing in structures on nanoscale level. Great step in this direction was successful application of nonlocal continuum field theory of Eringen. In the present paper, the free transverse vibration analysis is carried out for the system composed of multiple single walled carbon nanotubes (MSWCNT) embedded in a polymer matrix and under the influence of an axial magnetic field. Equivalent nonlocal model of MSWCNT is adopted as viscoelastically coupled multi-nanobeam system (MNBS) under the influence of longitudinal magnetic field. Governing equations of motion are derived using the Newton second low and nonlocal Rayleigh beam theory, which take into account small-scale effects, the effect of nanobeam angular acceleration, internal damping and Maxwell relation. Explicit expressions for complex natural frequency are derived based on the method of separation of variables and trigonometric method for the “Clamped-Chain” system. In addition, an analytical method is proposed in order to obtain asymptotic damped natural frequency and the critical damping ratio, which are independent of boundary conditions and a number of nanobeams in MNBS. The validity of obtained results is confirmed by comparing the results obtained for complex frequencies via trigonometric method with the results obtained by using numerical methods. The influence of the longitudinal magnetic field on the free vibration response of viscoelastically coupled MNBS is discussed in detail. In addition, numerical results are presented to point out the effects of the nonlocal parameter, internal damping, and parameters of viscoelastic medium on complex natural frequencies of the system. The results demonstrate the efficiency of the suggested methodology to find the closed form solutions for the free vibration response of multiple nanostructure systems under the influence of magnetic field.