The Securities and Exchange Commission (SEC) has previously monitored the online activity of the SpaceX and Tesla boss over concerns that his tweets could impact share prices and other markets.
Rumours of the latest investigation were first reported by financial newswire service First Squawk, who cited sources familiar with the matter.
“I hope they do! It would be awesome,” Mr Musk tweeted on Thursday.
When questioned by one of his 48 million Twitter followers why he tweeted about dogecoin so much, Mr Musk replied: “I love dogs and memes.”
In a 2018 interview with Bloomberg, Mr Musk said he “[does] not respect the SEC”, but would comply with the law if ordered to.
The Independent has contacted the SEC for comment.
The price of dogecoin dropped nearly 10 per cent after the report emerged, though this may have been the result of a wider market trend that saw other cryptocurrencies like bitcoin and Ethereum (ether) lose value in similar increments overnight.
Mr Musk has frequently shared images and memes relating to dogecoin in recent weeks, most recently on 24 February when he shared a mocked up image of a shiba inu dog in a spacesuit planting a dogecoin flag on the moon.
The cryptocurrency’s price often appears to move following endorsements from the technology billionaire, who once changed his Twitter profile to read, “CEO of Dogecoin”.
In a recent question-and-answer session on the Clubhouse app, Mr Musk said that his posts were “just meant to be jokes”, but added that “fate loves irony”.
He said: “The most entertaining outcome is often the most likely. Arguably the most entertaining outcome, and most ironic outcome, would be that dogecoin becomes the currency of Earth in the future.”
He also revealed that he bought dogecoin for his baby Lil X earlier this month, though it is not clear how much he invested.
These comments appeared to have had a positive impact on dogecoin’s price, but its notorious volatility means such spikes are often followed by a crash.
A similar but less pronounced pattern also appears to occur when Mr Musk tweets about bitcoin.
“Although Elon’s comments are clearly having an effect on crypto, this is only short-term,” Philippe Bekhazi, chief executive of cryptocurrency platform Stablehouse, told The Independent.
“We do not believe one person can have a long-term effect on bitcoin. It has been proven many times in the past that that is indeed the case.”