A furring strip fastening member is formed by a furring strip receiving plate, a bolt hole portion, a spring portion, and a rear face plate. The furring strip receiving plate and the rear face plate are shaped as flat plates that face each other. The bolt hole portion is provided at a central portion of the furring strip receiving plate. The spring portion connects the furring strip receiving plate and the rear face plate. The rear face plate has in a central portion thereof a through-hole. In a construction structure for buildings, a bolt is mounted to an anchor that is driven into a skeleton wall of a building that is formed by a concrete wall and a heat insulating material disposed outside the latter. This bolt traverses a through-hole in the rear face plate, and the bolt is screwed into a bolt hole portion. The furring strip is connected to the furring strip fastening member, and an external wall material is fastened to the furring strip. With the furring strip fastening member, the furring strip receiving plate presses against the furring strip while the rear face plate presses against the heat insulating material, on account of the reaction of the spring portion.