When executives at office space provider WeWork considered expanding into China two years ago, they discussed buying a fledgling Chinese co-working firm called UrWork. A deal didn’t get done, in part because WeWork offered stock, which UrWork didn't want to accept, a UrWork adviser said.

That was due to WeWork's rich stock value. UrWork executives felt their company had more room to grow. For WeWork, this was a missed opportunity: Today, UrWork is the largest co-working company in China, valued at more than $1 billion, and has begun to expand to the United States and other countries. Competition between the two has escalated. Last month, WeWork won a legal fight preventing UrWork from using its name on a soon-to-launch New York office.