"We wouldn't be against him having a different role," James Anderson, head of global equities for Baillie Gifford, told Barron's. "I don't think he needs to be CEO."
Baillie Gifford is the second-largest stakeholder in Tesla behind Musk. The U.K.-based investment firm owns 7.7 percent of the company — or roughly $3.8 billion worth — according to FactSet. Musk owns 19.7 percent, while the third-largest investor, T. Rowe Price, owns 5.2 percent.
Musk is facing contempt of court allegations after he tweeted what the SEC claims was "inaccurate" information about Tesla's production outlook. Tesla admitted to the SEC that Musk had not received approval to tweet the information, which is required by a settlement agreement struck between the agency and Tesla last fall.
Tesla has long touted the importance of keeping Musk at the helm of the company, saying in annual SEC filings that Tesla is "highly dependent on the services of Elon Musk, our Chief Executive Officer and largest stockholder."
Anderson agreed that Musk is essential to Tesla, but suggested a less formal role to Barron's like "chief ideologue."
In December, Baillie Gifford contributed to a $500 million funding round in Musk's rocket company, SpaceX, according to the Wall Street Journal. The firm is also an investor in another electric vehicle maker, NIO.
If Musk steps out of the CEO role, or is forced out of it, he would be walking away from potentially tens of billions of dollars over the next several years.
His compensation plan, approved by Tesla shareholders in March 2018, allows Musk to earn billions of dollars in stock grants as long as he remains CEO and hits certain milestones with the company. As an alternative, the plan allowed Musk to move into a chief product officer role while remaining executive chairman at Tesla. But his settlement with the SEC last year required Musk to give up his chairman role for at least three years.
Read more at Barron's.
— CNBC's Lora Kolodny contributed to this report.