Tesla CEO Elon Musk should not step down, but the board should really consider regulating Musk's Twitter use and appointing Musk a "No. 2," Jim Press, former COO and president of Toyota Motor Sales U.S.A., told CNBC on Monday.

"Twitter helps instantly communicate, so there are some advantages to it, but it needs to be disciplined and needs to be throttled in, I think," Press, who was also the deputy CEO of Chrysler, said on CNBC's "Closing Bell."

Just weeks after an Aug. 7 tweet in which Musk said he was considering taking Tesla private at $420 per share and had "funding secured," he abandoned the idea. In a Friday blog post, Musk explained that "it's apparent most of Tesla's existing shareholders believe we are better off as a public company."

But that may not be enough to dissuade the Securities and Exchange Commission, which, according to a New York Times report on Aug. 15, served Tesla with a subpoena to determine whether Musk violated securities laws by claiming he had funding for the take-private maneuver.

Furthermore, Musk himself has been criticized for erratic behavior, both online and off. He confessed in an interview with The New York Times the toll of the "excruciating" year he has had leading Tesla, particularly when crunching to meet Model 3 production goals.

"Twitter has become this direct line of public communication," Press said. "And in a good company with governance, you've got coordination, and you're working through a communications strategy within the organization. It seems to me that [Tesla's Twitter activity] needs to go through a more formal channel to be part of exactly what the company is saying."

Vocal Tesla short Gabe Hoffman, who founded activist hedge fund Accipiter Capital Management, thinks Musk's go-private tweet was more than just a slip-up in corporate communications.

"This was, in my view, the most egregious and naked example of securities fraud I have ever seen from a CEO in my 18 years as a hedge-fund manager," Hoffman said Monday on "Closing Bell."

"I believe imminently, in the next couple of months, Elon Musk will be removed as CEO of Tesla," Hoffman added.

William G. Pietersen, businessman and professor at Columbia University Graduate School of Business, placed some of the blame for the behavior of an "overwhelmed and emotionally exhausted" Musk on Tesla's board.

"If you think about a board's responsibility of oversight — to make sure there's a strong strategy in place to lead the company to a good place, that there's a team that's able to implement that strategy and to be able to ... insist that the CEO, in the interest of shareholders, undertakes those steps — I think there's a fundamental governance issue of responsibility here," Pietersen said in a "Closing Bell" interview.

But critical of the situation as Pietersen was, he said the frazzled CEO doesn't necessarily need to leave the company. He should really rethink how the company is being run, Pietersen said.

"The truth of the matter is this classic dilemma of whether the original visionary can lead a large, complex organization with a lot of operating problems," Pietersen said. "Leadership is a team sport; it's not a solo effort. And I think if you show me a successful company, I'll show you a strong leader that can lead a strong team."

Press, who just last week defended Musk's vision, agreed and maintained that Musk was still the right one to lead Tesla, so long as he had a team underneath him.

"He's really the visionary; he's the creative genius. His imprint is really on the company, but he has to stay high enough to stay strategic and let somebody else worry about the day-to-day business. A 'No. 2' makes a lot of sense," Press said.

"It seems logical as this company grows to bring in somebody to worry about growing the company, operating it day-to-day, and allowing him to really focus on strategy," he added.

Tesla closed down 1.10 percent at $319.27. Tesla is down about 8.3 percent year over year.