Neither companies responded to Reuters' requests for comment.

A deal could be seen as a response to the price wars that have disrupted the industry. Startups like Robinhood are quickly grabbing market share by eliminating commissions on stock trades. That prompted Schwab to scrap commissions on online trading of stocks, ETFs, and options. TD Ameritrade and other rivals quickly followed suit.

Also pressuring the discount brokers: low interest rates. That reduces the amount they earn off customer balances.

News of a potential deal drove TD Ameritrade shares up 25% at the market open Thursday. Schwab shares rose nearly 12%.