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Like Twitter, Tesla is a publicly traded firm with a turbulent and highly scrutinized stock. Like Square, SpaceX is a private firm (though Square has tentative plans to file for an IPO) at the cusp of a fast-growing industry.

The committee searching for a permanent CEO for Twitter said it will only consider candidates who can make a "full-time commitment." That would discount Dorsey, who vowed to keep his post at Square. He has repeatedly dodged questions about whether he would, if asked, remain at Twitter's helm.

But on Friday, billionaire investor Chris Sacca -- a highly influential venture capitalist with a big stake in Twitter -- urged Dorsey to stay on as permanent CEO of both companies.

In a series of tweets, the venture capitalist cast his ideal lineup for Twitter's c-suite: Dorsey as CEO, President of Global Revenues Adam Bain as chief operating officer and co-founder Ev Williams as executive chairman of the board.