WASHINGTON–Key senators were close to a deal on legislation to overhaul financial regulations, people familiar with the matter said, bringing the U.S. a step closer to sweeping changes to the way banks interact with consumers and the markets alike.

Top senators from each party were near a breakthrough agreement to create a new consumer-protection division within the Federal Reserve. This has been a contentious point due to heavy criticism of the Fed's past handling of its consumer-protection powers.

Senators...