The Securities and Exchange Commission is preparing to propose major changes to the stock market’s plumbing as soon as this fall.

Chair Gary Gensler directed SEC staff last year to explore ways to make the stock market more efficient for small investors and public companies. While aspects of the effort are in varying stages of development, one idea that has gained traction is to require brokerages to send most individual investors’ orders to be routed into auctions where trading firms compete to execute them, people familiar with the matter said.