  Based on first-principles calculations, we predict that the superconducting
MgB$_2$ with a AlB$_2$-type centrosymmetric lattice host the so-called phononic
topological Weyl nodal lines (PTWNLs) on its bulk phonon spectrum. These PTWNLs
can be viewed as countless Weyl points (WPs) closely aligned along the straight
lines in the $-$H-K-H direction within the three-dimensional Brillouin zone
(BZ). Their topological non-trivial natures are confirmed by the calculated
Berry curvature distributions on the planes perpendicular to these lines. These
lines are highly unique, because they exactly locate at the high-symmetry
boundary of the BZ protected by the mirror symmetry and, simultaneously,
straightly transverse the whole BZ, in different from known classifications
including nodal rings, nodal chains or nets, and nodal loops. On the
(10$\bar{1}$0) crystal surface, the PTWNLs-induced drumhead-like non-trivial
surface states appear within the rectangular area confined by the projected
lines of the PTWNLs with opposite chirality. Moreover, when the mirror symmetry
is broken, the double-degenerate PTWNLs are further lifted to form a pair WPs
with opposite chirality. Our results pave the ways for future experimental
study on topological phonons on MgB$_2$ and highlights similar results in a
series of isostructural AlB$_2$-type metallic diborides.
