An analytic approach is utilized to investigate the heat transfer mechanism in nanofluids containing spherical nanoelements. The transient thermal conduction process in a cylindrical geometry is assessed according to the Dual-Phase-Lag (DuPhlag) model and then compared with the classic Fourier solution. For the first several seconds, the wire temperature values calculated by the two models as well as their corresponding slopes are different; the curves merge at longer times depending on the nanofluid properties. At the initial stages, the realistic temperature magnitude of the suspension is also smaller than the Fourier solution, since in reality, the transient heat transfer rate is comparatively smaller owing to the certain time lags in the heat flux/temperature gradient relationship. The key parameters that govern in the initial stages of the heat transport inside the domain are the thermophysical properties of the components of a nanofluid, their relative velocity, the size/shape and concentration of the nanoparticles, and the system geometry. Two typical parameters, the conductivity and heat flux indexes, are also introduced to evaluate the deviation of the Fourier theory from the DuPhlag model. The conductivity index has a minimum value at the initial stages, and then, it is beginning to recover. A fluctuating distribution of the heat flux index is observed at the beginning of the process which is smoothed out after a lapse of time. A decrease in the nanoparticle diameter and/or an increase in the concentration may result in accelerating the transient response of the medium to the input thermal excitation.