Musk said the carmaker approached Sandberg two years ago for the Tesla COO position. Plus, the eccentric billionaire is the head of at least two other companies, including the rocket company Space X.
Tesla produced slightly more than 100,000 vehicles past year and the company has been struggling to boost production of its Model 3, which is less expensive than its first models and could help expand Tesla's base.
The newspaper reported that during an hour-long telephone interview Mr Musk alternated between laughter and tears, acknowledging he was working up to 120 hours a week and sometimes took Ambien to get to sleep.
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If Tesla directors are anxious about Musk - and they should be - here is advice for them: Do your jobs. "Is there someone who can do the job better?"
Company insiders told the paper that Musk's use of the drug was causing concern.
Tesla's board showed no sign of taking any action Friday.
"There have been many false and irresponsible rumors in the press about the discussions of the Tesla Board", Tesla said in a statement. Shortly after that Tesla stock surged almost 11 percent from $342 to $379 per share.
Erik Gordon, a University of MI business and law professor, told The Associated Press that Tesla's board now has a fiduciary duty to shareholders to take action.
The board has stood behind Mr Musk despite some weird behaviour.
"It was excruciating", he said, of the past 12 months, in which he berated analysts for asking "boneheaded" questions at an earnings conference call, and accused a British diver who helped rescue a Thai football team trapped underground of being a paedophile.
He also denied "being on weed" at the time of the tweet and said he saw it as an attempt at transparency, but admitted in the interview on Thursday that no one had seen or reviewed it before he posted it. Observers puzzled over the price he cited.
"If your plan is to take this company private, out of the public eye and the regulatory eye of the SEC, then people who have a fiduciary responsibility to their investors have to feel comfortable about the situation, and the New York Times article makes that harder", said Mike O'Rourke, chief market strategist at JonesTrading.
Tesla shares fell 3.6 percent to $323 in early trading.
One of the key questions Musk faced during the interview was whether he regretted his controversial decision to tweet last week about having the funding in place to take Tesla private.
"They're not dumb guys, but they're not supersmart", Musk told the Times.
The step, which had been widely-covered by global media, was expected to evoke even more troubles for the electric vehicles manufacturer if it became clear that Musk meant to temporarily boost the company's stock to force losses on short sellers. But Tesla isn't quite normal. Musk owns about 20 percent of the company.
Gordon said the board has to act now or be open to shareholder lawsuits. "They're at a point where they're going to have to distance their oversight from any prior relations".
Musk's admissionscome at a tumultuous time for Tesla, but the CEO predicted that the pain isn't quite over yet.
"Without the cult of personality around Elon, they're just a cult of personality that's burning a lot of cash".