This article provides the domain of existence Ω of the Laplace transform of infinitely divisible negative multinomial distributions. We define a negative multinomial distribution on Nⁿ, where N is the set of nonnegative integers, by its probability generating function which will be of the form (A(a₁z₁,…,a_{n}z_{n})/A(a₁,…,a_{n}))^{-λ} where A(z)=∑_{T⊂{1,2,…,n}}a_{T}∏_{i∈T}z_{i}, where a_{∅}≠0, and where λ is a positive number. Finding couples (A,λ) for which we obtain a probability generating function is a difficult problem. Necessary and sufficient conditions on the coefficients a_{T} of A for which we obtain a probability generating function for any positive number λ are know by (Bernardoff, 2003). Thus we obtain necessary and sufficient conditions on a=(a₁,…,a_{n}) so that a=(e^{t₁},…,e^{t_{n}}) with t=(t₁,…,t_{n}) belonging to Ω. This makes it possible to construct all the infinitely divisible multinomial distributions on Nⁿ. We give examples of construction in dimensions 2 and 3.