  Near equilibrium, Green-Kubo relations provide microscopic expressions for
macroscopic transport coefficients in terms of equilibrium correlation
functions. At their core, they are based on the intimate relationship between
response and fluctuations as embodied by the equilibrium
fluctuation-dissipation theorem, a connection generically broken far from
equilibrium. In this work, we identify a class of perturbations whose response
around far-from-equilibrium steady states is linked to steady-state correlation
functions via an equilibrium-like fluctuation-response equality. We then
utilize this prediction to substantiate linearized hydrodynamic transport
equations that describe how spatial inhomogeneities in macroscopic
nonequilibrium systems relax. As a consequence, we derive nonequilibrium
Green-Kubo relations for the transport coefficients of two types of
hydrodynamic variables: local conserved densities and broken-symmetry modes. A
byproduct of this work is to provide a theoretical foundation for the validity
of Onsager's regression hypothesis around nonequilibrium steady states. Our
predictions are analytically and numerically corroborated for two model
systems: density diffusion in a fluid of soft, spherical active Brownian
particles and phase diffusion in the noisy Kuramoto model on a square lattice.
