Let D = (V, A) be a finite simple directed graph (digraph). A function f : V −→ {−1, 1} is called a twin signed k-dominating function (TSkDF) if f (N − [v]) ≥ k and f (N + [v]) ≥ k for each vertex v ∈ V. The twin signed k-domination number of D is γ * sk (D) = min{ω(f) | f is a TSkDF of D}. In this paper, we initiate the study of twin signed k-domination in digraphs and present some bounds on γ * sk (D) in terms of the order, size and maximum and minimum indegrees and outdegrees, generalising some of the existing bounds for the twin signed domination numbers in digraphs and the signed k-domination numbers in graphs. In addition, we determine the twin signed k-domination numbers of some classes of digraphs.