  In this paper, we study the asymptotic behavior of solutions to the
three-dimensional incompressible Navier-Stokes equations (NSE) with periodic
boundary conditions and potential body forces. In particular, we prove that the
Foias-Saut asymptotic expansion for the regular solutions of the NSE in fact
holds in {\textit{all Gevrey classes}}. This strengthens the previous result
obtained in Sobolev spaces by Foias-Saut. By using the Gevrey-norm technique of
Foias-Temam, the proof of our improved result simplifies the original argument
of Foias-Saut, thereby, increasing its adaptability to other dissipative
systems. Moreover, the expansion is extended to all Leray-Hopf weak solutions.
