  Recently, Dragani\'c, Munh\'a Correia, Sudakov and Yuster showed that every
tournament on $(2+o(1))k^2$ vertices contains a $1$-subdivision of a transitive
tournament on $k$ vertices, which is tight up to a constant factor. We prove a
counterpart of their result for immersions. Let $f(k)$ be the smallest integer
such that any tournament on at least $f(k)$ vertices must contain a
$1$-immersion of a transitive tournament on $k$ vertices. We show that
$f(k)=O(k)$, which is clearly tight up to a multiplicative factor. If one
insists in finding an immersion of a complete directed graph on $k$ vertices
then an extra condition on the tournament is necessary. Indeed, we show that
every tournament with minimum out-degree at least $Ck$ must contain a
$2$-immersion of a complete digraph on $k$ vertices. This is again tight up to
the value of $C$ and tight on the order of the paths in the immersion.
