Ordinary homotopy invariants do not take care of the behavior of spaces at infinity. So proper homotopy invariants are needed to distinguish non-compact spaces. The proper Lusternik-Schnirelmann category of a space $X$ is the minimum $n$ such that $X$ can be covered by $n+1$ closed subspaces which can be properly deformed to a ray in $X$. This proper numerical invariant was introduced by Ayala, DomÌnguez, Márquez and Quintero in 1992. They characterized $\mathbb{R}^n$  as the unique $n$-manifold with 1 strong end and proper L-S category 2. In dimension 3, a result of Gómez-Larrañaga and  González-Acuñastates that the ordinary L-S category of a closed 3-manifold is characterized by its fundamental group. As a first step in studying the relationship between proper L-S category and the fundamental pro-group of an open 3-manifold, we show that a large class of Whitehead manifolds has proper L-S category 3, the maximum possible. This is a sharp contrast with ordinary L-S category, since all these 3-manifolds are contractible. The proper L-S category can not be characterized by fibrations or or fat wedges because those constructions do not exist in general in proper homotopy theory. However the former computation is carried out by following the ideas of Ganea, but changing in some sense fibrations by cofibrations.