A report in the Daily Telegraph today quotes a source close to founders Biz Stone and Jack Dorsey as saying the founders "are being approached with offers of wheelbarrow loads of cash".
However, it says that Stone and his partner are "being very careful about who they accept money from and who they give their equity to".
The service, which now has as many as 1.7m UK users and 7m globally is used heavily by those working in the media, digital and celebrities.
Most notably among the celebrities using it are Stephen Fry who has attracted more than 322,000 followers or those who subscribe to his 140 character long postings.
Fry's tweet on the death of Jade Goody has been widely quoted today: "I suppose she was a kind of Princess D from the wrong side of the tracks."
For Barack Obama it played a part in his digital communications campaign for the presidency.
Others stars using it include NBA Pheonix Suns basket ball star Shaquille O'Neal who recently Twittered during a break from a game.
Despite the offers of such cash Twitter is not ready to accept new investors having already turned down $500m from Facebook, which recently redesigned its homepage annoying some users, and looks increasingly like Twitter according to some.
Twitter says that the $35m it recently received from venture capitalist Benchmark and Institutional Venture Partners, which values the company at $250m, is sufficient.
Stone said that Twitter was not "actively seeking more funding" as it still has cash left from previous funding rounds when it accepted $15m of investment from Amazon chairman and chief executive Jeff Bezos's Bezos Expeditions and Spark Capital last summer.