A closed orientable 3-manifold admits such a Morse-Bott function exactly when it is a connected sum of lens spaces, copies of S2 × S1, and torus bundles over S1.
Gelbukh, Reeb Graphs of Morse-Bott Functions on a Given Surface , Bulletin of the Iranian Mathematical Society, Volume 50 Article number 84, 2024
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Characterizing $3$-dimensional manifolds represented as connected sums of Lens spaces, $S^2 \times S^1$, and torus bundles over the circle by certain Morse-Bott functions
A closed orientable 3-manifold admits such a Morse-Bott function exactly when it is a connected sum of lens spaces, copies of S2 × S1, and torus bundles over S1.