Social Media

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Twitter's incoming interim CEO says he is committed to maintaining the company's strategy, no matter the criticism it has received from the outside.

Outgoing Twitter CEO Dick Costolo and his interim replacement, chairman and co-founder Jack Dorsey, spoke with CNBC's "Squawk on the Street" on Friday. Dorsey said during that conversation he "believe(s) in" the social media firm's current strategy, but he hopes to accelerate some of its work.

"We're focused on our execution, we're focused on getting product out there, we're focused on getting services, and we have the right team in place to do that," he said. "And I do feel that we can accelerate a bunch of that so people get to see it faster."

The company announced last night that Costolo would step down on July 1, with Dorsey taking the helm while Twitter conducted a CEO search.

Jack Dorsey, CEO of Twitter.
John Chiala | CNBC

Many investors and analysts have called for a shakeup amid a sluggish run for Twitter's stock. The company has struggled to expand its user base and grow revenue through advertising and other streams. Investors initially reacted positively to Costolo's departure Thursday, as shares spiked as much as 9 percent in extended trading.

By Friday, however, the stock was trading at less than 3 percent above the prior day's close.

Part of that response may have been due to Dorsey's insistence on a Thursday evening call that Costolo was not forced out, and that he did not anticipate any major strategic changes.

Read MoreAfter CEO exit, Twitter says no strategy change

"The fundamentals...that the company is building right now and executing right now are extremely strong and beautiful," Dorsey said Friday.

Sources confirmed to CNBC that Costolo left of his own volition, and that it was a move he had been considering for some time.

"This transition is not the result of anything more than Dick deciding to move on from his role as CEO," Dorsey said on the Thursday call. "There is no connection with our near-term results."

Explaining the timing of that decision, Dorsey said it came down to keeping business running smoothly.

"We did not want to interrupt that momentum, if anything we want to amplify it because we're extremely excited about the products and new initiatives to come," Dorsey said. He explained that this momentum can be seen in new products like Periscope.

Despite Dorsey's insistence to the contrary, many maintain that Twitter is in need of a big shift—and that change may need to come to its board.

"We're seeing the consequences of poor governance here...Twitter is rapidly becoming the BlackBerry of social media," Harvard Business School's Bill George said.

Discussion among analysts and some in the media has turned to whether Twitter could potential be sold to a larger tech firm. Costolo, however, said Friday that the board is not looking for a sale.

"Jack and I and the rest of the board all know that we have a fiduciary duty to our shareholders to maximize value, and we think we can do that best as an independent company," he said, declining to comment on "the existence or non-existence" on any conversations with other firms.

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Dorsey, who also leads financial services company Square, will be splitting his time between the two firms. Some sources have suggested to CNBC that Dorsey will be in contention for taking the permanent Twitter chief position, but he told CNBC that's not a priority for him at the moment.

"It's not even a question I'm considering right now," he said, responding to a second question about whether he would take the CEO job by saying he's "not focused on it."

Still, not everyone is convinced that Dorsey would make a good permanent CEO for the company.

"I can understand putting [Dorsey] in that seat for a time so they can take their time and make a good decision," said Kevin Landis, Firsthand Capital Management's CEO and CIO. "The way he answered that question about whether he would take that job permanently, I think there were reasons for why he stepped down from that job the first time, and those reasons are probably still valid."

—CNBC's Jacob Pramuk and David Faber contributed to this report.