Let $C$ be a 2-category and $Mnd(C)$ the 2-category of monads in $C$. As explained by Street in [the formal theory of monads](https://doi.org/10.1016/0022-4049(72)90019-9), the Eilenberg-Moore construction is right 2-adjoint to the inclusion 2-functor $C \to Mnd(C)$ sending an object to the identity monad on it. Therefore it preserves 2-limits.

A [fibration in a 2-category](https://ncatlab.org/nlab/show/fibration+in+a+2-category) may be defined in terms of certain 2-limits (namely slice categories) and adjoints, and so is preserved by any 2-functor preserving 2-limits, and in particular by the Eilenberg-Moore construction.

Thus it's sufficient for $p$ to be a fibration object in the 2-category $Mnd(C)$.