The Obama administration appears set to chart a new course for U.S. space exploration by promoting the use of private companies to ferry astronauts into orbit, according to people familiar with the matter.

A Soyuz spacecraft is set to launch Monday in Kazakhstan, with a Russian flight commander and U.S. and Japanese flight engineers.The controversial plan would mark a trailblazing departure for the nation's space program by allowing a group of closely held start-up companies, for the first time, to compete for a central role...