It is unclear whether Square is profitable and what its revenue is. The company has at times encountered bumps with products that have failed, such as Square Wallet, a way to pay for goods without taking out a credit card. Companies with less than $1 billion in annual revenue can file for an I.P.O. without publicly disclosing the paperwork until closer to the time they are ready to begin a road show for investors.

Image Square Stand enabled iPads to accept credit card payments. The company has expanded beyond payments into lending. Credit... Jim Wilson/The New York Times

The I.P.O. filing is not yet public. Square did disclose some details of its ownership in February in a document filed with the Department of Commerce in Alaska. According to the filing, Mr. Dorsey owns 26.2 percent of Square, making him the largest shareholder by far. The next biggest investor is Khosla Ventures, a venture capital firm, with 17.4 percent. Others with big stakes include Mr. McKelvey, the co-founder and a director, with 9.4 percent.

There’s little chance underwriters and auditors will let Mr. Dorsey serve as the chief executive of two publicly traded companies, according to investment bankers. That would appear to diminish the likelihood of Mr. Dorsey taking on the role of permanent chief of Twitter, which he co-founded in 2006 and where he has been chairman since 2008.

Yet while Twitter’s board has been busy interviewing outside candidates for the chief executive job, some Twitter directors still hope that Mr. Dorsey will change his mind and decide he wants to keep the job, according to people close to the situation. In June, Twitter’s board issued a statement saying it would consider only candidates who were in a position to make a full-time commitment to Twitter.

If Mr. Dorsey does remain committed to Square, he could retain the chairman’s role at Twitter and advise the new chief executive.

So far, Mr. Dorsey has been plunging into his interim chief role at Twitter. He has been meeting with various teams to familiarize himself with their products and plans, even as the company has continued to push out a steady stream of product changes, from eliminating the background wallpaper that people see in their desktop Twitter feeds to new tools for marketers to better target their ads.