This paper gives a construction of noncommutative Poisson algebras. Such algebras are widely used in noncommutative geometry and mathematical physics. The authors start with the classification of all the inner Poisson structures on a finite-dimensional path algebra \(kQ\) using the decomposition into decomposable Lie ideals of the standard Poisson structure on \(kQ\). All the finite quivers \(Q\) without oriented cycles such that \(kQ\) admits outer Poisson structures are then determined. The authors prove that these quivers are exactly the finite quivers without oriented cycles such that there exist two non trivial paths lying in a reduced closed walk, which cannot be connected by a sequence of non trivial paths.