At J-PARC (Japan Proton Accelerator Research Complex), the COMET (Coherent Muon to Electron Transition) experiment is being prepared. Pion-capture and muon-transport superconducting solenoid magnets will be used to capture pions and transport muons which are produced by pion decays. Since the radiation level is high near the magnets, a cold box and a current lead box, which supply a two-phase forced helium flow and currents to the magnets, will be located apart from the magnets. To connect the cold box and current lead box to the magnets, unique cryogenic transfer tubes that have thermal joints for installing NbTi superconducting lead wires at the insides of the transfer tubes are necessary. The transfer tubes should be able to make the wires thermally stable. In this paper, thermal contact conductance between three round-shaped NbTi/Cu monolith superconducting wires and a rectangular slot in a copper block is measured from 4.5 K to 10 K in six cases and a concept of a thermal joint is proposed for the unique transfer tube of the muon-transport superconducting solenoid magnet. In the thermal joint, the round-shaped superconducting wires make line contacts with the slot covered with one layer of a Kapton tape. The void volume in the slot is filled with Apiezon N grease. An experimental apparatus that uses liquid helium is utilized for measuring the thermal contact conductance in a sample of the thermal joint. The effect of thermal cycles on the thermal contact conductance is observed. The measured thermal contact conductance is used to determine the length of the thermal joint and the size of the gap between the thermal joints. (C) 2016 Elsevier Ltd. All rights reserved.