Referenews

It isn’t clear who would take over leading Twitter if Mr. Musk steps aside as CEO, or what his role would be, given that he still owns the company. Most of the company’s prior leadership was either fired or left after he took over.

“No one wants the job who can actually keep Twitter alive. There is no successor,” Mr. Musk tweeted Sunday.

Twitter didn’t respond to a question on the poll’s outcome.

The billionaire’s leadership at Twitter has been tumultuous and weighed on sentiment toward his other businesses, particularly car maker

Tesla.

Shares in the auto maker have fallen more than 56% this year, frustrating some retail investors who partly blame Mr. Musk’s focus on Twitter for the decline. Mr. Musk’s Twitter involvement also has dented Tesla’s brand image.

Tesla shares initially advanced Monday following the suggestion that Mr. Musk would stop running Twitter, though later fell along with the broader market.

There have been questions since Mr. Musk first showed interest in buying Twitter how he would juggle running the company while also pursuing all his other endeavors.

Bill Nelson,

the National Aeronautics and Space Administration’s administrator, this month said he had asked SpaceX President

Gwynne Shotwell

if Twitter would divert from the rocket company’s mission. “She assured me that it would not be a distraction,” Mr. Nelson said.

Mr. Musk said last month that he had too much work on his plate.

“I expect to reduce my time at Twitter and find somebody else to run Twitter over time,” he testified at a trial about his Tesla pay package last month. He also said he has been spending most of his time of late focusing on Twitter. At the time, he wrote on Twitter, “I will continue to run Twitter until it is in a strong place, which will take some time.”

The outcome of the poll adds another disruption at the top of Twitter, which has suffered waves of leadership upheaval. Co-founder and former CEO

Jack Dorsey

handed over running the company little over a year ago after a bruising battle with an activist investor. His successor,

Parag Agrawal,

lasted less than a year and was fired by Mr. Musk immediately after he took over the company in October. Mr. Musk also ousted several other top Twitter executives, and more have resigned since.

Mr. Musk has been the driving force at Twitter for nearly two dramatic months, with many advertisers having paused spending, resignations, and content-moderation actions that have been abrupt and controversial among many users. This past week has been another one full of twists. Twitter suspended the accounts of several journalists, prompting criticism from some lawmakers and officials before Mr. Musk said Saturday that Twitter would begin reinstating the accounts.

Earlier Sunday, Twitter made another sudden change to its content-moderation rules, saying it would no longer allow “free promotion of certain social media platforms.” That also fueled criticism and questions, including from Mr. Dorsey, who previously endorsed Mr. Musk’s takeover. “Doesn’t make sense,” Mr. Dorsey tweeted Sunday.

Late Sunday, the tweets from @TwitterSupport and a note on the company’s website announcing the policy against promoting other social-media platforms appeared to have been deleted. Twitter’s head of trust and safety,

Ella Irwin,

said Monday that the company removed the new policy “in response to user feedback.” Ms. Irwin said that links to other social-media sites will be allowed until Twitter determines whether or not to implement a new policy prohibiting them.

Mr. Musk, who has said he intends to make the platform a bastion of free speech, appeared to acknowledge some concerns about the recent, abrupt content policy changes. “Going forward, there will be a vote for major policy changes. My apologies. Won’t happen again,” he tweeted shortly before posting the poll about his leadership Sunday.

Mr. Musk has warned about Twitter’s financial condition, saying last month that the company had suffered “a massive drop in revenue” and was losing over $4 million a day. He later raised the possibility of bankruptcy.

Mr. Musk’s team this past week reached out for potential fresh investment for Twitter at the same price as the original $44 billion deal, a Twitter investor said.

Some Tesla shareholders have complained that Mr. Musk’s recent focus on Twitter is hurting the auto maker. This month he sold more than $3.5 billion of Tesla stock in his second round of sales since buying Twitter. It wasn’t clear what prompted Mr. Musk to sell additional Tesla stock.

Democrat Sen. Elizabeth Warren of Massachusetts on Sunday sent a letter to Tesla’s board chair,

Robyn Denholm,

raising questions about Mr. Musk’s involvement with Twitter, suggesting it could be to the detriment of the auto maker’s shareholders and create conflicts of interest. The senator has sparred before with Mr. Musk.

Tesla didn’t respond to a request for comment about the share sale or the senator’s letter.

—Peter Stiff contributed to this article.

Write to Alexa Corse at alexa.corse@wsj.com