In this work, the effect of multi-walled carbon nanotube (CNT) and montmorillonite nanoclay on polymer chain dynamics is investigated around the percolation concentration for systems based on ethylene vinyl acetate (EVA) copolymer. Then, the results obtained are compared with literature data to determine if, regardless of particle characteristics, a universal rheological behavior can be detected at percolation. To do so, rheological analyses are performed under small amplitude oscillatory shear (SAOS), large amplitude oscillatory shear (LAOS), and transient shear step. SAOS data showed that, while the dynamics related to the Rouse relaxation time (tau (R)) were not significantly influenced, the reptation relaxation time (tau (D)) was strongly increased by the presence of nanoparticles. In step shear transient tests, the critical shear rate for overshoot appearance was decreased due to chain confinement, and the formation of particle network strongly increased the level of stress overshoot. Particle networks increased significantly the nonlinear parameters (I (3)/I (1) and Q (0)) obtained under LAOS and quantified by FT-rheology. In all measurements, due to the higher surface area associated to its size and density as well as hollow structure, CNT showed stronger effects compared to clay. Moreover, while the percolation concentration was different for CNT and clay, both systems showed similar behavior at percolation: a 0.5 scaling for G' indicating a Rouse-dominated behavior.