The red-hot rumor about "Twoogle"—the purchase of Twitter by Google—seems not to be true, but it's stirring up questions about the future of Twitter and what, exactly, it might do with Google.

This all started with a post on TechCrunch late on Thursday claiming Google was about to buy Twitter.

Since Twitter these days is one of the most talked-about properties on the Web—getting attention from Stephen Colbert, senators, sports figures—the rumor has raced around the Web.

By this morning, insiders were denying the story. In fact, it seems unlikely Twitter would consider selling to anyone yet. (Neither Twitter or Google responded to requests for on-the-record comment.)

Twitter co-founder Biz Stone appeared on The Colbert Report just last night, noting that the Twitter team wants to try to build their phenomenon into a "strong, profitable independent company."

In an interview excerpted in the March issue of Condé Nast Portfolio, Twitter CEO Evan Williams proclaimed his intention to go it alone.

"We feel like we're just getting started," Williams said. "To sell it now—even at a really good price—would seem a little bit disappointing because it could be so much more and I want to see what we can make it."

While a full-on merger is unlikely, Twitter and Google may be discussing a partnership. Williams has ties to Google—he founded Blogger, which he sold to Google in 2003—and the two companies could help each other in key ways.

Twitter is a way to search what's happening on the Web in real time. Its search function pops up the latest "tweets" on any topic, and continues to add new results as new tweets on the topic are posted. So it can act as kind of a rolling ticker showing what people are saying about anything right at that moment. Google has no way to do that now.

(This morning, a search on Twitter for "google twitter" was rolling out more than 100 new posts a minute. Notably, it turned up a tweet from Revolution CEO Steve Case saying, "Deal or no deal? My bet: no deal; at least not right now.")

At the same time, Twitter is having trouble making the most of its search function. Williams told Portfolio that the search button on Twitter is hard to find for a reason: Twitter's systems can't yet handle massive numbers of search queries. A big search company like Google or Microsoft could help Twitter bring its real-time search to the forefront.

Williams is also aware that Google, Microsoft or Yahoo might try to create their own Twitter-like services, though he doesn't consider that a serious threat.

"I'm pretty sure they all are about to launch something like this, but we can't pay too much attention to that," Williams said. "My time in Google taught me that there's no one better than Google at implementing Internet products, but focus is a really big deal.

"If you're in Google," Williams added, "you could be working on the next big thing but it's not as important as search and advertising, and it never will be."