The Cash app, which launched in 2015, is growing faster than PayPal 's Venmo, according to Nomura Instinet. Since early 2016, Square Cash app downloads have averaged 128 percent year-over-year growth each month versus Venmo's 74 percent growth, according to the Nomura analysis.

The company had 7 million active customers on its money-transfer app in December alone, Square said in its recent quarterly letter to shareholders.

"People are using this as their primary banking account, and in some case it's their only bank account," Dorsey said on stage at the Consensus blockchain conference in New York this week. "We are reaching an audience that is under-served and even to the point of unbanked, which wasn't a stated goal but it's something we love and want to lean into more."

Square customers are treating the payment company's Cash app more like a bank account than the company intended, CEO Jack Dorsey said this week. While that wasn't a goal, he said Square plans to capitalize on the trend.

"The number one and two merchants we see in the Cash app are Walmart and McDonalds," said Dorsey, who is also the CEO of Twitter. "This is everyday usage."

Nomura analyst Dan Dolev told CNBC the banking opportunity is "huge" for Square, and that likely isn't priced to the stock.

"Square could become a one-stop shop," Dolev said. "Think about a world where you have a Square bank account but also it's the merchant, so you're basically you're able to get around a lot of the fees the banking industry is currently charging."

The fin-tech company signaled its plans to break into banking last year.

Square filed for an FDIC licence to be an industrial loan company in Utah, American Banker first reported in September. The license would allow it to perform some of the same services as banks, including issuing loans without having rely on a bank to originate them, and without having to get licenses in each state.

Square launched cryptocurrency trading through its Cash app in January, which Dolev said is another main reason the app has grown in popularity.

Dorsey said on the conference panel Wednesday he's a "huge fan" of bitcoin and its underlying technology. Dorsey also said he wants Square to help its long-term adoption.

"The internet is going to have a native currency, so let's not wait for it to happen, let's help it happen," Dorsey said. "I don't know if it will be bitcoin, but I hope it will be."

Shares of Square slipped as much as 3 percent this week after news that PayPal would pay $2.2 billion to buy iZettle, which provides mobile card readers. The deal could be a threat to Square, which also sells its payment hardware to small businesses. The stock later recovered, trading near $55 Friday.