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Given a symmetric operad $\mathcal{P}$ and a $\mathcal{P}$-algebra $V$, theassociative universal enveloping algebra ${\mathsf{U}_{\mathcal{P}}}$ is anassociative algebra whose category of modules is isomorphic to the abeliancategory of $V$-modules. We study the notion of PBW property for universalenveloping algebras over an operad.  In case $\mathcal{P}$ is Koszul a criterion for the PBW property is found. Anecessary condition on the Hilbert series for $\mathcal{P}$ is discovered.Moreover, given any symmetric operad $\mathcal{P}$, together with a Gr\"obnerbasis $G$, a condition is given in terms of the structure of the underlyingtrees associated with leading monomials of $G$, sufficient for the PBW propertyto hold.  Examples are provided.

PBW property for associative universal enveloping algebras over an operad