McCue has previously been a Twitter board member, who stepped down due to the increasing competition between Flipboard and the service he co-founded. With a reputation as a product genius in Silicon Valley, he would also be an obvious heir apparent to CEO Dick Costolo, who has been under siege from investors over the last six months.
Flipboard, which is backed by some of Silicon Valley's most prominent venture capitalists with over $160 million in investment, has its own growth problems. Last year, it claimed it had 100 million "activated" users, which is those people who have used the service some of the time. In its last investment round, it was valued at about $800 million and sources said it has about $50 million in revenue.
Read MorePiggybacking Icahn, other activists beats market
But people familiar with the company say its user numbers have leveled off recently, in part because its app is no longer automatically installed in Samsung phones. It's also just increasingly hard to attract audiences and some think that even as good a news reader as Flipboard has a limited appeal.
Buying Flipboard doesn't automatically help Twitter increase its own user base of 302 million people. And it's logical to assume that many Flipboard users are also Twitter users, since both services can appeal to voracious digital media consumers.
Twitter seems to have figured out how to sell ads against this base, while Flipboard has been in a much more experimental mode. Last year, it added video ads and more recently rolled out "promoted items," similar to what Twitter has long offered.
Still, especially due to McCue, Flipboard is an attractive acquisition target and other interested buyers might include Google and perhaps Facebook.
Reps for both Twitter and Flipboard declined to comment.
—By Kara Swisher, Re/code.net.
CNBC's parent NBC Universal is an investor in Re/code's parent Revere Digital, and the companies have a content-sharing arrangement.