The move would be a new effort by Twitter to get back into music after a failed attempt last year to launch a music-finding service.
The report suggested that such a deal, if consummated, would be expensive for Twitter, but would add a company with a customer base of 250 million, roughly the same size as that of the messaging platform.
SoundCloud recently raised some $60 million in venture capital, which would give the start-up a value of some $700 million according to The Wall Street Journal.
Neither company has commented on the report.
Re/code said the deal could appeal to SoundCloud by giving the service easier ability to deliver revenue-generating ads along with its music streams.