The report was largely welcomed by investors, and Twitter's stock rose briefly by as much as 6.8 percent, before closing 5.3 percent higher at $26.94 on the New York Stock Exchange.
But the news raised questions over whether Dorsey could run both companies at the same time, and whether that would create a conflict of interest.
"Jack Dorsey would be the right guy (for the job) if he were not running Square," said FBN Securities analyst Shebly Seyrafi. "They (Square) have a lot of challenges competing with a lot of other payments companies, and that's going to need a lot of attention."
Some investors said Dorsey was a more effective leader now than in 2008, when the co-founder was fired from his first stint as Twitter CEO. Press reports suggested Dorsey's attention was on other projects, and he struggled to manage frequent outages during a period of exploding growth.
"No one knows the product better than Jack," said SunTrust Robinson Humphrey analyst Robert Peck, citing faster product rollouts since Dorsey took over the top post on an interim basis on July 1. "You can see his fingerprints on the company already."
Twitter declined to comment on the Re/code report.
Dorsey has not dismissed the idea of becoming permanent CEO of Twitter while staying at Square, but Twitter's board has said its next CEO must be focused solely on that company.
Square, which pioneered the use of instant payments over smartphones and is worth about $6 billion based on its most recent round of funding, plans to file for an initial public offering shortly, according to a source familiar with the situation.
Dorsey's success with Square gives him renewed clout to lead Twitter, said Gartner analyst Brian Blau.
"He's already invented a whole new category," said Blau. "That gives him a lot of credibility when it comes to what to do next with Twitter."
Twitter's last permanent CEO, Dick Costolo, announced his decision to step down in June, under mounting concerns about the company's slow user growth and seeming inability to attract advertisers at the same rate as competitors.
Costolo is now set to leave Twitter's board, Re/code reported.
Other potential candidates mentioned for the top job at Twitter include Adam Bain, the company's president and head of revenue; Instagram founder Kevin Systrom; and Evan Williams, another co-founder of Twitter who was once its CEO and heads blogging website Medium.
© Thomson Reuters 2015
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