Is composing homotopies of spheres bilinear?

Question

$\newcommand{\SS}{\mathbb{S}}$ For nonnegative integers $n, m$ and $k$ consider the map $$ c: \pi_n(\SS^m) \times \pi_m(\SS^k) \to \pi_n(\SS^k) $$ given by composition, i.e. $([f], [g]) \mapsto [g \circ f] = g_*[f]$. $g_*$ is a homomorphism $\pi_n(\SS^m) \to \pi_n(\SS^k)$, so $c$ is $\mathbb{Z}$-linear (additive) in $[f]$. Is this also additive in in $[g]$?