For a graph G=(V,E) with a vertex set V and an edge set E, a function f:V→{0,1,2,...,diam(G)} is called a broadcast on G. For each vertex u∈V, if there exists a vertex v in G (possibly, u=v) such that f(v)>0 and d(u,v)≤f(v), then f is called a dominating broadcast on G. The cost of the dominating broadcast f is the quantity ∑v∈Vf(v). The minimum cost of a dominating broadcast is the broadcast domination number of G, denoted by γb(G). A multipacking is a set S⊆V in a graph G=(V,E) such that for every vertex v∈V and for every integer r≥1, the ball of radius r around v contains at most r vertices of S, that is, there are at most r vertices in S at a distance at most r from v in G. The multipacking number of G is the maximum cardinality of a multipacking of G and is denoted by mp(G). It is known that mp(G)≤γb(G) and that γb(G)≤2mp(G)+3 for any graph G, and it was shown that γb(G)−mp(G) can be arbitrarily large for connected graphs (as there exist infinitely many connected graphs G where γb(G)/mp(G)=4/3 with mp(G) arbitrarily large). For strongly chordal graphs, it is known that mp(G)=γb(G) always holds. We show that, for any connected chordal graph G, γb(G)≤⌈32mp(G)⌉. We also show that γb(G)−mp(G) can be arbitrarily large for connected chordal graphs by constructing an infinite family of connected chordal graphs such that the ratio γb(G)/mp(G)=10/9, with mp(G) arbitrarily large. This result shows that, for chordal graphs, we cannot improve the bound γb(G)≤⌈32mp(G)⌉ to a bound in the form γb(G)≤c1⋅mp(G)+c2, for any constant c1<10/9 and c2.