Let Y be a smooth complex projective variety of dimension N + 1 endowed with a very ample invertible sheaf O(1), let d ∈ ℕ^{∗}, let V^{d} ⊂ H^{0}(Y, O(d)) be the locus of smooth hypersurfaces of class c_{1}(O(d)), let F ∈ V^{d} and let j_{F} : X_{F} → Y denote the closed embedding. By Lefschetz theory the monodromy action of π_{1}(V^{d}, F) on H^{N}(X_{F} , ℂ)/j^{∗}H^{N}(Y, ℂ) is irreducible. Let W ⊂ Y be a closed subscheme and let V^{d}(W) ⊂ V^{d} be the space of smooth hypersurfaces containing W. We show that if the ideal of W is spanned in degree small with respect to d then for all F ∈ V^{d}(W) the monodromy action of π_{1}(V^{d}(W), F) is irreducible on the orthogonal (for the cup-product) of the projections on H^{N}(X_{F} , ℂ)/j^{∗}H^{N}(Y, ℂ) of the classes of N-dimensionnal irreducible components of W. The proof is essentially topological ; difficulties arise from the fact that W may have any singularities.