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"I talk constantly internally about the impact, the platform has on the world and the way we're growing the business," Costolo said in an exclusive "Squawk on the Street" interview. "We're excited about the business we've got and we like running it as an independent company."
The deal would make Twitter more searchable, expand the number of people who see tweets, or what Twitter calls its "logged-out user base." It is expected the company would make money from a licensing fee rather than advertising.
This is not the first time the two online companies have worked together. Between 2009 and 2011, Twitter and Google had a similar deal to provide Google with Twitter data more efficiently and comprehensively, but the deal was not renewed.
The new tie-up quickly sparked speculation that the renewed relationship would blossom into a marriage.
"If Twitter is smart, they'll sell their business now to Google," Ross Gerber, CEO of wealth-management firm Gerber Kawasaki, told CNBC. "Get out of the business now while you're at the top."
But Costolo said the deal was part of its "total audience strategy," including trying to tap users who aren't logged in to Twitter. He noted it would be several months before the service would be fleshed out.
Costolo has made a big push in recent quarters to shift attention away from the slowing growth of its core user base—284 million at the end of the third quarter—to the full range of all people who see tweets on any platform.
Twitter is also preparing to tackle another issue that has dented its ability to add new users: the trolls.
Costolo said he aims "to make Twitter and all these kinds of platforms a clean, well lit place … without harassment or abuse."
Earlier this week, an internal memo from Costolo was leaked to the media. "We suck at dealing with abuse and trolls on the platform and we've sucked at it for years," the memo read. "We lose core user after core user by not addressing simple trolling issues that they face every day," it said.
Costolo told CNBC that Twitter plans to take action quickly.
"[You want to] change the economic equation around [the abuse] so the onus and burden isn't on the person being abused to report it, but rather the onus and the burden are on people who are engaging in harassment to prove that they should be allowed to continue to use the platform," Costolo told CNBC.
"It should just be taken care of automatically and algorithmically by the platform and we have a number of initiatives underway to make that the case."
Twitter has faced a number of high-profile trolling cases in the recent past. In January, Anita Sarkeesian, a feminist commentator on sexism in video games, recently published a week's worth of messages she received via Twitter, including a large number of obscenity-laden rape and death threats.
Interview by Carl Quintanilla. Julia Boorstin contributed to this article.
—By CNBC.Com's Leslie Shaffer; Follow her on Twitter