Elon Musk had framed the potential sale as a way to cover any taxes he would need to pay if lawmakers imposed new taxes on unrealized capital gains. He began selling billions of dollars worth of stock a few days after his tweet.

Kimbal Musk sold 88,500 shares one day before the Tesla CEO tweeted about the potential sales of his own. The company’s shares fell sharply in the wake of Mr. Musk’s poll—58% of voters said he should sell—indicating the tweet was viewed as negative news.

Spokesmen for the SEC and Tesla didn’t respond to a request to comment.

Tesla has recently accused the SEC of harassing the company and its chief executive by repeatedly launching new enforcement investigations. The friction dates to a 2018 lawsuit in which regulators accused Mr. Musk of misleading investors with a tweet that said he could take the company private and had the funding to do so.

One question for regulators, according to securities lawyers, would be whether Mr. Musk told his brother about his upcoming tweet or about the timing of his sales before Kimbal Musk traded on Nov. 5—or if Kimbal Musk otherwise learned of the poll and then traded. Kimbal Musk serves on Tesla’s board of directors.

Kimbal Musk’s trading could violate rules that generally prohibit employees and board members from trading on material nonpublic information. Employees and directors of public companies generally can’t buy or sell shares when they are aware of undisclosed material information.

Insider trading law historically covers the misuse or theft of information that belongs to a public company. Mr. Musk’s plans to trade, however, arguably belonged to him, and not to Tesla and its shareholders, said Adam Pritchard, a law professor at the University of Michigan. In that case, it is unclear whether insider trading prohibitions would apply to Mr. Musk’s actions or to his brother’s trades, Mr. Pritchard said.

“What follows is that if the SEC wanted to pursue this, it would be a hard-fought question in court,” Mr. Pritchard said. “If they decided to pursue it, Elon Musk would be willing to spend a little bit to pursue it in court.”

Mr. Musk told The Financial Times Thursday that Kimbal Musk was unaware he planned to conduct the Twitter poll, while the company’s lawyers did know about it.

Officers and directors can avoid insider-trading charges when they buy or sell under a preset trading plan. Using such a program, known as a 10b5-1 plan, insiders can trade at predetermined intervals provided they don’t change the plan when they have material nonpublic information.

Kimbal Musk has often traded Tesla stock under a 10b5-1 plan, according to securities filings. He has made more than 40 disclosures since 2011 indicating the sales were made under a plan, according to regulatory filings.

But a disclosure filed with the SEC on Nov. 5 didn’t say that he used such a program for those trades. The filing said he donated 25,000 shares to charity in addition to selling 88,500 shares. After the sale, he still owned 511,240 shares.

Kimbal Musk’s second-biggest sale of Tesla shares was in February 2021, when he sold $25 million of stock at around $852 a share—a price the stock wouldn’t reach again until October. The disclosure for that sale also didn’t say Kimbal Musk used a 10b5-1 plan for those trades.

Elon Musk had created his own 10b5-1 plan to sell Tesla shares on Sept. 14, according to a filing that disclosed his November trades. The disclosure said the trades were “automatically effected” pursuant to the plan and involved stock options that had vested and were scheduled to expire this year.

The price of Tesla’s shares began declining after Mr. Musk began selling on Nov. 8 and have fallen 33% since then.

The SEC’s civil investigation, which began last year, could end without the regulator making formal allegations of wrongdoing. The SEC sometimes closes probes without taking enforcement action.

Mr. Musk, Tesla and the SEC have been feuding since 2018, when the CEO landed in trouble over a tweet that regulators said was misleading. Mr. Musk wrote that he had “funding secured” to take Tesla private at $420 a share. After an investigation, the SEC said that Mr. Musk had never discussed such a going-private deal and that his statement, which caused Tesla’s stock to skyrocket, constituted fraud.

Mr. Musk and Tesla settled the SEC lawsuit in 2018 by each agreeing to pay $20 million, and Mr. Musk stepped down as chairman. He also agreed to preclear tweets that were deemed material to Tesla shareholders with Tesla’s lawyers. A Manhattan federal court approved the policy, which covered a list of events, financial metrics and product announcements.

Despite the settlement, Mr. Musk has maintained a combative posture with the SEC. A lawyer for Tesla and Mr. Musk last week accused the SEC of harassing Mr. Musk with “serial investigations” of him and the company. The claim was made in a letter that the attorney,

Alex Spiro,

filed with a federal judge who oversaw the 2018 settlement.

In another letter to the court, Mr. Spiro alleged the SEC had leaked information about an investigation. On Thursday, U.S. District Judge

Alison Nathan

issued an order, writing that she couldn’t force the SEC to respond to the claim because the letter “does not contain specific facts or legal authority to justify this request.”

The SEC has repeatedly questioned whether Mr. Musk violated the pact by tweeting about Tesla production numbers, the company’s stock price, and other matters. SEC attorneys wrote Tesla in 2019 and 2020 asking why certain tweets weren’t cleared by company lawyers. Tesla pushed back, saying the statements weren’t covered by the court judgment.

Mr. Musk tweeted on Wednesday: “I didn’t start the fight, but I will finish it.”

—Mark Maremont, Liz Hoffman and Rob Copeland contributed to this article.

Write to Dave Michaels at dave.michaels@wsj.com