The people tasked with overseeing Elon Musk’s plans for Tesla Inc. —its board of directors—have received solid support from shareholders over the years but criticism from some investors and advocates who say they lack independence.

Boards have enormous responsibility in corporate deals, especially ones as complex and fraught as the buyout of Tesla that Mr. Musk suggested this week. Most of Tesla’s directors have close business or personal relationships with Mr. Musk that they would have to balance against their obligation to ensure that any deal serves the interests of Tesla shareholders beyond its famous leader, corporate governance specialists say.

The board’s role in the possible buyout was clouded by Mr. Musk’s unusual way of announcing the idea—in a sudden, very brief tweet on Tuesday. That tweet was followed more than 20 hours later by a short statement from six directors saying the board had met several times since Mr. Musk told it of his go-private idea last week, and that it was “taking the appropriate next steps to evaluate this.”

The sequence of events suggests that “the board review has been very, very informal,” said Adam Epstein, who heads corporate-governance consultant Third Creek Advisors.

Mr. Musk’s announcement attracted scrutiny from the Securities and Exchange Commission, which has asked Tesla whether Mr. Musk was truthful when he said in his tweet that he had secured funding for the buyout.