James Murdoch is the board director of the electric-car maker.
Murdoch was testifying in a trial regarding a shareholder lawsuit against Musk over the CEO's compensation package.
Murdoch said that Musk actually has but only in the last few months.
Insider had a request for comment from aTesla spokesman, but he didn't reply.
Murdoch said in the deposition that Musk never said he was leaving or reducing his involvement in the company. Murdoch made more information available during his testimony.
Murdoch said they had conversations during the difficult times. Conversation about his exhaustion, but no explicit threats.
Murdoch didn't offer any clues as to who the successor might have been.
In the last few weeks, the billionaire has been working a lot at the micro-blogging site. Musk said during his testimony on Wednesday that he will soon be able to focus on other things.
"After acquisition, there is an initial burst of activity needed to reorganize the company, but then I will reduce my time commitment to find someone else to run the company over time," Musk said.
Musk said during his testimony that he doesn't want to be the CEO of any company and that he's more of an engineer than a businessman.
Musk said that his CEO role at the two companies is more about him than it is about the CEO. The technology in the car that makes it successful is my responsibility.
Musk is being sued over his compensation package. Musk and the automaker were accused of breaching their fiduciary duties by giving Musk a pay package that was beyond the bounds of reasonable judgement. The package was necessary to keep Musk focused on the company.