Dorsey still has the full support of the Twitter board, sources said, because of its continued belief in Dorsey as a product person. Dorsey is fully focused on improving Twitter's user experience, the sources said.
Shares of Twitter are down more than 21 percent so far this year in choppy trade, amid reports the company might consider selling itself . But rumored suitors have reportedly pulled back. In October,Salesforce.com ruled out a bid.
"Our stockholders heard that we were involved in a process and they made it very clear that they did not want us to buy that company," Salesforce CEO Marc Benioff told CNBC late last month . "So, very specifically, we had to walk away ... I love the CEO. I love the company. I love the brand. I love everything about Twitter but our stockholders don't and I listen to them."
Dorsey was named Twitter's permanent CEO last fall after a lengthy search process. The company has seen slower growth compared with rival platforms like Facebook and Snapchat, and has looked toward live video as a way to bring users and advertisers to the site.
Meanwhile, Square is " firing on all cylinders ," as the company pushes its newer reader which accepts chip cards and near-field-communication payment methods like Apple Pay.
It's rare to run two publicly traded companies at once — late Apple co-founder and CEO Steve Jobs once struggled with the obligations of running Pixar and Apple, according to his official biography.
"I think it's virtually impossible to run two companies. And both are very, very different and both are very, very challenged. And that's not to say there's anything wrong with Twitter," former interim Yahoo CEO Ross Levinsohn told CNBC last month.
Twitter did not immediately respond to a request for comment.
Correction: Shares of Twitter have fallen more than 21 percent so far this year. An earlier version misstated the percentage.
— CNBC's Michelle Fox, Elizabeth Gurdus and Christine Wang contributed to this report.
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