A closed-loop cooling concept for 21 T Fourier transform ion cyclotron resonance (FT-ICR) superconducting magnets is presented. In the magnet system, low temperature superconducting coils are immersed in a subcooled 1.8 K bath, which is connected to the saturated helium reservoir through the weight load relief valve. Saturated liquid helium is refrigerated by a Joule-Thomson (JT) heat exchanger and flows through the JT valve, isenthalpically dropping its pressure to approximately 1.6 kPa, corresponding to a saturation temperature of 1.8 K. Helium gas exhausted from JT pump is liquefied by a two-stage cryocooler located after the vapor purify system. In the present paper, the amount of heat budget is determined and the structural design of cryostat is carried out by the relevant analyses. The position of a cryocooler in the magnet system is investigated, taking into account the requirement of magnetic field for normal performance. Helium liquefaction system, a key component of the closed-loop cooling system, is fabricated and tested in order to demonstrate the feasibility of our new cryogenic cooling for high field magnets.