Recent gravitational-wave observations are reporting a number of merger events of BHs with several 10 solar masses. Possible origins of such systems are of great interest, and close binaries with massive stars are often considered as potential progenitors. An interesting possibility is that massive stellar binaries born in the early universe evolve into massive BH-BH binaries, that eventually merge in the nearby universe after a long delay time. However, there is no established picture on how such postulated stellar binaries could form. In this talk I focus on the so-called primordial star formation, where there is no heavy elements contained in the gas. I first review the standard theoretical framework, and then investigate how possible (or difficult) is the "close and massive" binary formation in the earliest universe.
