We consider the double-dimer model in the upper-half plane discretized by the square lattice with mesh size $δ$. For each point $x$ in the upper half-plane, we consider the random variable $N_δ(x)$ given by the number of the double-dimer loops surrounding this point. We prove that the normalized fluctuations of $N_δ(x)$ for a fixed $x$ are asymptotically Gaussian as $δ\to 0+$. Further, we prove that the double-dimer nesting field $N_δ(\cdot) - \mathbb{E}\, N_δ(\cdot)$, viewed as a random distribution in the upper half-plane, converges as $δ\to 0+$ to the nesting field of CLE(4) constructed by Miller, Watson and Wilson.