A bloc of WeWork directors is planning to push Adam Neumann to step down as chief executive after a tumultuous week in which his eccentric behavior and drug use came to light, and the startup delayed its much-anticipated stock-market listing.

A group including officials tied to SoftBank Group Corp., the company’s largest investor, wants Mr. Neumann to relinquish his title of CEO of We Co., the parent of the office-sharing company, people familiar with the matter said.

The...