The multi-billionaire will be its chief executive after buying the company last week, ending months of back and forth over the $44bn (£38.3bn) deal.
He has moved quickly to put his mark on the firm, which is used by politicians and journalists around the world.
The reforms he is contemplating include changes for how Twitter verifies accounts, as well as job cuts.
The Washington Post has reported that a first round of cuts is under discussion that could affect 25% of the company's staff.
Twitter did not respond to a request for comment from the BBC on the report.
Top executives have already been removed, as Mr Musk brings in high profile allies to the company.
Twitter co-founder Jack Dorsey has rolled his entire stake of 18m shares, worth almost $978m at the buyout price of $54.20, into the new private company, according to a filing with the US Securities and Exchange Commission on Monday.
Mr Dorsey, who left Twitter's board in May, supported Mr Musk's purchase of the firm.
"Elon is the singular solution I trust. I trust his mission to extend the light of consciousness," he said in a tweet after the takeover was approved by the company's former management team.
Meanwhile, technology investor Jason Calacanis who changed his Twitter bio to "chief meme officer", said he was "hanging out at Twitter a bit ... during the transition".
He took to the social media site to solicit opinions on a range of topics from advertising to video.
Mr Calacanis also asked how much people would pay to be verified, amid reports that Twitter could charge people $20 per month to keep the blue ticks that indicate verified accounts.
Later on Tuesday, responding to a Twitter user calling for the platform to purge accounts that had been inactive for more than a year, Mr Musk said "definitely".
Mr Musk, who leads electric car maker Tesla, has changed his biography on the social media site to "Chief Twit".
He is now the sole director of Twitter, another filing on Monday showed.
The nine ousted directors include former chairman of the board Bret Taylor and former chief executive Parag Agrawal.
