Many of the stock market’s biggest players have taken a hands-off approach to bitcoin and other cryptocurrencies. That could start to change.

Names familiar to the trading world, including retail brokerage TD Ameritrade Holding Corp. and high-speed trading giant Virtu Financial Inc., are backing an exchange called ErisX, which plans to trade cryptocurrencies and related derivatives, according to the companies. ErisX unveiled its new business plan Wednesday.

Separately,...