We consider k-quasi-M-hyponormal operators T ∈ B(ℋ) such that TX = XS for some X ∈ B K ℋ and prove a Fuglede–Putnam-type theorem when the adjoint of S ∈ B K is either a k-quasi-M-hyponormal or a dominant operator. We also show that two quasisimilar k-quasi-M-hyponormal operators have identical essential spectra.