We construct the most general four-dimensional \( \mathcal{N} \) = 4 supergravity coupled to an arbitrary number n of vector multiplets in which the global scaling symmetry is gauged, in addition to a subgroup of SL(2, ℝ) × SO(6, n). The various gaugings are parametrized by an embedding tensor built out of \( 2\left(\begin{array}{c}n+6\\ {}3\end{array}\right)+4\left(n+6\right) \) parameters that satisfy a specific set of quadratic consistency constraints, to which we provide explicit solutions. We also derive the local supersymmetry transformation rules and the equations of motion for the four-dimensional \( \mathcal{N} \) = 4 matter-coupled supergravity with local scaling symmetry. Such supergravity theories do not possess an action, since the scaling symmetry is only an on-shell symmetry of the corresponding ungauged theories.
