A modified hydrogenated diene polymer composition comprising: a modified hydrogenated diene polymer obtained by hydrogenating a diene polymer having a vinyl bond content of 20 to 70%, the modified hydrogenated diene polymer having a weight average molecular weight (Mw) of 100,000 to 1,700,000, a molecular weight distribution (Mw/Mn) of 1.0 to 3.0, and a hydrogenation rate of 72 to 96%, and containing one or more functional groups; wherein the number average molecular weight (Mn) and the weight average molecular weight (Mw) are determined by gel permeation chromatography at 130°C; wherein the vinyl bond content is calculated by the Hampton method using infrared absorption spectrometry; and wherein the hydrogenation rate is calculated from a 1 H-NMR spectrum; and a diene rubber, the modified hydrogenated diene polymer composition containing the modified hydrogenated diene polymer and the diene rubber in an amount of 100 parts by mass in total, the amount of the modified hydrogenated diene polymer being 60 to 95 parts by mass, and the amount of the diene rubber being 5 to 40 parts by mass, wherein the functional groups introduced into the component are chosen among an alkoxy group and an amino group, and the functional groups have been introduced into the component by reacting a polymerization terminator containing functional groups with the active site of the diene polymer, and wherein the number of functional groups contained in a single molecular chain of the modified hydrogenated diene polymer is 0.1 to 5.0; and wherein the number of functional groups contained in a single molecular chain is a value determined by titration as defined in the description. a modified hydrogenated diene polymer obtained by hydrogenating a diene polymer having a vinyl bond content of 20 to 70%, the modified hydrogenated diene polymer having a weight average molecular weight (Mw) of 100,000 to 1,700,000, a molecular weight distribution (Mw/Mn) of 1.0 to 3.0, and a hydrogenation rate of 72 to 96%, and containing one or more functional groups; wherein the number average molecular weight (Mn) and the weight average molecular weight (Mw) are determined by gel permeation chromatography at 130°C; wherein the vinyl bond content is calculated by the Hampton method using infrared absorption spectrometry; and wherein the hydrogenation rate is calculated from a 1 H-NMR spectrum; and wherein the number average molecular weight (Mn) and the weight average molecular weight (Mw) are determined by gel permeation chromatography at 130°C; wherein the vinyl bond content is calculated by the Hampton method using infrared absorption spectrometry; and wherein the hydrogenation rate is calculated from a 1 H-NMR spectrum; and a diene rubber, the modified hydrogenated diene polymer composition containing the modified hydrogenated diene polymer and the diene rubber in an amount of 100 parts by mass in total, the amount of the modified hydrogenated diene polymer being 60 to 95 parts by mass, and the amount of the diene rubber being 5 to 40 parts by mass, wherein the functional groups introduced into the component are chosen among an alkoxy group and an amino group, and the functional groups have been introduced into the component by reacting a polymerization terminator containing functional groups with the active site of the diene polymer, and wherein the number of functional groups contained in a single molecular chain of the modified hydrogenated diene polymer is 0.1 to 5.0; and wherein the number of functional groups contained in a single molecular chain is a value determined by titration as defined in the description. the modified hydrogenated diene polymer composition containing the modified hydrogenated diene polymer and the diene rubber in an amount of 100 parts by mass in total, the amount of the modified hydrogenated diene polymer being 60 to 95 parts by mass, and the amount of the diene rubber being 5 to 40 parts by mass, wherein the functional groups introduced into the component are chosen among an alkoxy group and an amino group, and the functional groups have been introduced into the component by reacting a polymerization terminator containing functional groups with the active site of the diene polymer, and wherein the number of functional groups contained in a single molecular chain of the modified hydrogenated diene polymer is 0.1 to 5.0; and wherein the number of functional groups contained in a single molecular chain is a value determined by titration as defined in the description.