The Wall Street Journal has claimed that that Jack Dorsey's social media site has told the video platform's Chinese owner, ByteDance, they are keen to make an offer.
It comes after US President Donald Trump threatened to ban TikTok in the US from September 15, if the firm is not bought by Microsoft or another American company by then.
However, sources have told The WSJ that Twitter may not have the cash to buy TikTok and complete on a deal in a 45-day window.
Trump has said he believes the US government should get a huge lump sum from the TikTok sale.
After Microsoft confirmed they are in talks to buy the Chinese-owned video app, the President said the Treasury should benefit from a "substantial portion" of the money made from a sale of the social media platform if it's sold to a US firm.
Trump has accused TikTok of giving the Chinese government private data.
And he has demanded: "The United States should get a very large percentage of that price, because we're making it possible.
"It would come from the sale, which nobody else would be thinking about but me, but that's the way I think, and I think it's very fair."
The world leader had said that TikTok's US operations "will close down on September 15 unless Microsoft or somebody else is able to buy it and work out a deal, an appropriate deal so the Treasury ... of the United States gets a lot of money."
TikTok has profusely denied the app is a security threat.
The firm - which has a new CEO in ex-Disney chairman Kevin Meyer - no longer operates in Hong Kong, after the national security law was placed in the power of the Chinese government.