Tesla has been searching for a number-two executive to help take some pressure off Chairman and CEO Elon Musk, The New York Times reported on Thursday.

Executives at the electric car-maker have been trying to recruit a chief operating officer or another number-two executive for years to take over some of Musk's day-to-day responsibilities and that search intensified following his recent headline-making tweets about taking the company private, the Times said, citing sources.

Tesla did not immediately respond to CNBC's emailed requests for comments. A statement provided to The New York Times attributed to the company's board excluding Musk said, "There have been many false and irresponsible rumors in the press about the discussions of the Tesla board ... We would like to make clear that Elon's commitment and dedication to Tesla is obvious."

Last week, Musk tweeted that he was "considering taking Tesla private at $420. Funding secured." Tesla shares rose following the tweet but the CEO's statement that he had funding secured came under scrutiny. Musk later explained that his claim was based on repeated and ongoing conversations with the Saudis and tweeted the names of financial and legal advisors he said he was working with on the proposal to take Tesla private.

Elon Musk Tweet: Am considering taking Tesla private at $420. Funding secured.

Multiple news outlets reported on Wednesday that the Securities and Exchange Commission had served Tesla with a subpoena following Musk's tweets. A subpoena would be one of the first steps in a formal inquiry. On Thursday, the Wall Street Journal reported that the SEC is looking into whether Musk tweeted about taking Tesla private to hurt those who were shorting the company's stock.

The Times' reporting about the search for a number-two executive was included in an article about an interview with Musk, during which the Tesla CEO talked about his intense workload and discussed the moments that lead up to the tweet. He also told the Times that Tesla board members had not complained to him about the go-private tweet.

Musk told the newspaper that Tesla had several years ago approached Facebook Chief Operating Officer Sheryl Sandberg for the number-two job, but he added that there was "no active search right now."

The Times said that sources told the newspaper some Tesla board members have expressed concern about both Musk's workload and his use of Ambien, a sedative. The newspaper reported, citing an unnamed source, that some board members have connected Musk's use of the medication to his after-hours tweeting. Multiple members of the company's board are also aware that Musk has occasionally used recreational drugs, the Times said, citing multiple people familiar with the matter.

For more on the newspaper's interview with Elon Musk, read the article from The New York Times.

— CNBC's Sara Salinas contributed to this report.