  In this paper, we prove propagation of $\frac{1}{s}$-Gevrey regularity $(s
\in (0, 1))$ and Analyticity $(s=1)$ for the Vlasov-Navier-Stokes system on
$\mathbb{T}^d \times \mathbb{R}^d$ (or $\mathbb{R}^d\times\mathbb{R}^d$) using
a Fourier space method in analogy to the results proved for the Euler system in
[KV09] and [LO97] and for Vlasov-Poisson system in [VR21]. More precisely, we
give quantitative estimates for the growth of the $\frac{1}{s}$-Gevrey norm and
decay of the regularity radius for the solution of the system in terms of the
force field $\nabla_x u$, the local density $\rho_f$ and the volume of the
support in the velocity variable of the distribution of particles $f$. In
particular, this implies existence of $\frac{1}{s}$-Gevrey $(s \in (0, 1))$ and
Analytic $(s = 1)$ solutions for the Vlasov-Navier-Stokes system in
$\mathbb{T}^d\times\mathbb{R}^d$ (or $\mathbb{R}^d\times\mathbb{R}^d$) and
global Gevrey solutions in $\mathbb{T}^3\times\mathbb{R}^3$.
