The stock was up 6.58 percent at $20.82 in midday trading on the New York Stock Exchange, marking the first time since early January that Twitter shares have topped $20.
The rise came after a New York Times story citing people familiar with the matter as saying that Twitter was exploring the potential of an application for the Apple TV device that allows streaming of online content to television screens.
San Francisco-based Twitter has been ramping up its live video offerings.
Twitter-owned Periscope lets people broadcast live streams of whatever they wish and the social network known for one-to-many messaging has been making a growing number of deals to stream events.
Twitter and the National Basketball Association last month announced a deal to stream live content, but not games, to users of the social media platform.
Earlier this year Twitter struck a deal with the National Football League to stream Thursday night American football games, and it also streamed content from the Wimbledon tennis tournament.
An agreement with Apple TV could extend Twitter video broadcasts to the biggest screens in homes, engaging more users and increasing the potential for revenue.
Online video is one of the latest initiatives of the social network to invigorate its platform in the face of stagnating user ranks and persistent financial losses.
Twitter stock has lost two thirds of its value since its historic peak of $73.31 dollars recorded in December 2013, a few weeks after making its debut on Wall Street at an IPO price of $26.