Digital environment involves numerous transformations for the media system. One of them is the process that is reconfiguring the traditional media visibility parameters from mass communication era. To study it, we use, taking an innovative approach, hyperlinks as proxy to the media visibility. Our goal is to find out the destination of the hyperlinks aimed at the media in the context of the political conversation on Twitter generated around an event of public relevance: the negotiations on the formation of the Spanish Government. The methodology is based on machine learning in a big data sample of 127.3 million tweets. This will allow us to make new contributions to how media is restructuring its social visibility as a result of integrating into the digital. The results indicate the prominence of pure players and social media platforms in the linking practices of users. By contrast, legacy media places in a secondary position. Also, geopolitical context plays a key role in conditioning the use of hyperlinks by Twitter users. Finally, our data reveal that the media have a limited capacity to determine the hyperlink network.