The SuperCritical Water-cooled Reactor (SCWR) is one of the candidates for the fourth-generation nuclear power plant, and it uses light water as a coolant. Heat transfer between a fuel assembly and water may degrade at certain conditions of supercritical pressure flows. Therefore, accurate and reliable estimation of heat transfer coefficients is necessary for the design of the fuel assembly and the reactor core. A series of heat transfer tests has been carried out at a test facility named SPHINX by using carbon dioxide as a stimulant of water. The tests produced heat transfer data of the supercritical pressure flows inside a circular tube of 4.4-mm inside diameter at varying operating pressures, mass fluxes, and wall heat fluxes. The test range of the mass flux was 400 to 1200 kg/m2 s, and the maximum heat flux was 150 kW/m2. The operating pressures were 7.75, 8.12, and 8.85 MPa in the tests. The test results were compared with estimations of the existing correlations for supercritical pressure flows. The comparison showed reasonable agreement between our data and the correlations. However, a rather large departure from the normal heat transfer correlations was observed near pseudocritical temperatures. Besides the comparison of the normal heat transfer coefficients, the onset of heat transfer deterioration was compared between the test cases and two existing criteria. One of the criteria was derived from experiments by using Freon but with a test section of identical geometry while the other criterion was for a flow of carbon dioxide in a larger bore channel than ours. Both criteria showed fair agreement with our experiment. Most test cases with noticeable heat transfer degradation were located in the region of deterioration predicted by the criteria.