The White House has decided to begin funding private companies to carry NASA astronauts into space, but the proposal faces major political and budget hurdles, according to people familiar with the matter.

The controversial proposal, expected to be included in the Obama administration's next budget, would open a new chapter in the U.S. space program. The goal is to set up a multiyear, multi-billion-dollar initiative allowing private firms, including some start-ups, to compete to build and operate spacecraft capable of ferrying...