We consider the Gaussian interface model in the presence of random external fields, that is the finite volume (random) Gibbs measure on $\mathbb{R}^{Λ_N}$, $Λ_N=[-N, N]^d\cap \mathbb{Z}^d$ with Hamiltonian $H_N(φ)= \frac{1}{4d}\sum\limits_{x\sim y}(φ(x)-φ(y))^2-\sum\limits_{x\in Λ_N}η(x)φ(x)$ and $0$-boundary conditions. $\{η(x)\}_{x\in \mathbb{Z}^d}$ is a family of i.i.d. symmetric random variables. We study how the typical maximal height of a random interface is modified by the addition of quenched bulk disorder. We show that the asymptotic behavior of the maximum changes depending on the tail behavior of the random variable $η(x)$ when $d\geq 5$. In particular, we identify the leading order asymptotics of the maximum.