  We obtain an upper bound for the Morse index of Willmore spheres
$\Sigma\subset S^3$ coming from an immersion of $S^2$. The quantization of
Willmore energy shows that there exists an integer $m$ such that
$\mathscr{W}(\Sigma)=4\pi m$. Then we show that
$\mathrm{Ind}_{\mathscr{W}}(\Sigma)\leq m$. The proof relies on an explicit
computation relating the second derivative of $\mathscr{W}$ for $\Sigma$ with
the Jacobi operator of the minimal surface in $\mathbb{R}^3$ it is the image of
by stereographic projection thanks of the fundamental classification of Robert
Bryant.
