We consider the problem of obtaining a list of compact 3-manifolds that can be obtained as a union of given sets. The concept of an A-category of a manifold Mⁿ as introduced in [CP] is a generalization of the Lusternik-Schnirelmann category cat(Mⁿ). For a fixed closed connected k-manifold A, 0 ≤ k ≤ n−1, a subset B in Mⁿ is said to be A-contractible if there are maps φ: B → A and α: A → Mⁿ such that the inclusion map i : B → M is homotopic to α⋅φ. The A-category cat_A(Mⁿ) of Mⁿ is the smallest number of sets, open and A-contractible needed to cover Mⁿ. Thus when A is a point P, cat_P (Mⁿ) = cat(Mⁿ). In the case A =S¹ it was shown in [GGH2] that the fundamental group of a closed 3-manifold M with cat_S¹ (M) = 2 is cyclic and it then follows from Perelman’s work [MT] that in this case M is a lens space; hence M can be covered by two open solid tori. As a first step to obtaining a list of all 3-manifolds with cat_S¹ (M) = 3 we ask about minimal covers of M by three open sets, each homotopy equivalent to S¹. In particular we consider covers of M by three open solid tori.