Elon Musk, the world’s richest man, reached out to Parag Agrawal, Twitter’s chief executive, a few weeks ago with a friendly heads-up. He was buying shares of the social media company, Musk confided, and wanted to discuss how to make Twitter better.

Musk had ideas for reshaping social networks that dovetailed with those of Agrawal and Jack Dorsey, Twitter’s co-founder, according to their public exchanges. All three have floated the notion of radically shifting the power in social networking to users and away from behemoth companies, by using an approach to technology that would give people control over what they see in their social media feeds.

In the ensuing weeks, Agrawal discussed having Musk become a more active participant in Twitter’s future, according to two people with knowledge of the conversations who were not authorized to speak publicly. Agrawal also welcomed having Musk — who has more than 80 million Twitter followers and sometimes tweets a dozen or more times a day — join the company’s board, one of the people said.