  We consider manifolds equipped with a foliation $\cal F$ of codimension $4q$,
and an almost quaternionic structure $Q$ on the transversal bundle of ${\cal
F}$. After discussing conditions of projectability and integrability of $Q$, we
study the transversal twistor space $Z{\cal F}$ which, by definition, consists
of the $Q$-compatible almost complex structures. We show that $Z{\cal F}$ can
be endowed with a lifted foliation ${\hat {\cal F}}$ and two natural almost
complex structures $J_1$, $J_2$ on the transversal bundle of $\hat{\cal F}$. We
establish the conditions which ensure the projectability of $J_1$ and $J_2$,
and the integrability of $J_{1}$ ($J_{2}$ is never integrable).
