Let (Mn, g) be a complete simply connected n-dimensional Riemannian manifold with curvature bounds Sectg g < κ for κ < 0 and Ricg g (n - 1)Kg for K < 0. We prove that for any bounded domain Ω ⊂ Mn with diameter d and Lipschitz boundary, if Ω∗ is a geodesic ball in the simply connected space form with constant sectional curvature κ enclosing the same volume as Ω, then σ1(Ω) < Cσ1(Ωz.ast;), where σ1(Ω) and σ1(Ω∗) denote the first nonzero Steklov eigenvalues of Ω and Ω∗ respectively, and C = C(n, κ,K, d) is an explicit constant. When κ = K, we have C = 1 and recover the Brock-Weinstock inequality, asserting that geodesic balls uniquely maximize the first nonzero Steklov eigenvalue among domains of the same volume, in Euclidean space and the hyperbolic space. © The authors. Published by EDP Sciences, SMAI 2025.