A three-dimensional integrable generalization of the Stäckel systems is proposed. A classification of such systems is obtained, which results in two families. The first family is the direct sum of the two-dimensional system which is equivalent to the representation of the Schottky-Manakov top in the quasi-Stäckel form and a Stäckel one-dimensional system. The second family is probably a new three-dimensional system. The system of hydrodynamic type, which we get from this family in the usual way, is a three-dimensional generalization of the Gibbons-Tsarev system. A generalization of the quasi-Stäckel systems to the case of any dimension is discussed.