The episode has sparked concerns even among some fans of the Tesla chief executive that he might risk stretching himself too thin. Mr. Musk also runs rocket company SpaceX and helped start a tunneling enterprise and a neuroscience startup working on brain-implant technologies. Now with Twitter, some Tesla supporters are concerned that he could lose focus.

“I wish he would walk away,” said Gary Black, managing partner of the Future Fund LLC, which owns roughly $50 million worth of Tesla, according to FactSet. Mr. Black, a Tesla booster, said he views Twitter as a distraction that is likely to demand more of Mr. Musk’s time than the billionaire expects.

A close follower of Tesla recently tweeted at Mr. Musk and the Twitter CEO: “Elon, Twitter is an unnecessary distraction. Just focus on Tesla.”

Mr. Musk has sought to quell such anxiety. “To be clear, I’m spending <5% (but actually) of my time on the Twitter acquisition. It ain’t rocket science!” he tweeted last week. “Tesla is on my mind 24/7.”