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Let Cn be the set of n complex vectors endowed with a norm ‖⋅‖Cn. Let A,B be two complex n×n matrices and τ a positive integer. In the present paper we consider the nonlinear difference equation with delay of the type uk+1=Auk+Buk−τ+Fk(uk,uk−τ),      k=0,1,2,…, where Fk:Cn×Cn→Cn satisfies the condition ‖Fk(x,y)‖Cn≤p‖x‖Cn+q‖y‖Cn,       k=0,1,2,…, where p and q are positive constants. In this paper, absolute stability conditions for this equation are established.