In June, India banned the app as well as a host of other Chinese products, part of escalating tensions between the two countries.
TikTok's international growth began when Beijing-based media and entertainment giant ByteDance bought Musical.ly, a Chinese company which had US operations, in 2017, and merged it with its own app the following year.
Last year it was the world's second-most downloaded app, behind Facebook's WhatsApp.
Microsoft declined to comment.
If Trump were to force through a sale, it is expected it would come through the US Committee on Foreign Investment in the United States, which has ordered companies including the dating app Grindr to be sold by its Chinese owners.
ByteDance's US investors, which include several large venture capital firms, are believed to have discussed buying out TikTok's US operations.
Buying TikTok would almost certainly be Microsoft's biggest acquisition to date. It paid US$26 billion (NZ$39b) for LinkedIn in 2016.