The new device won't arrive until the second half of next year, people familiar with the matter told Bloomberg.
While the Galaxy Round features a gently concave screen and body, the new phone would feature a three-sided screen that would allow users to view messages and other alerts from an angle. The three sides could also work independently of each other, according to the sources.
Samsung isn't commenting on the rumours, but a patent application filed by the company and published this week shows how the screen might work in practice.
According to the filing, one side of the curved display could be used to navigate up and down while the user reads on the main screen. It could also be used to lock the phone, or feature shortcuts to access specific files, folders or apps.
According to Bloomberg, the new phone will feature the same "Youm" OLED display tech used on the Galaxy Round.
It isn't clear whether Samsung will add the new device to its Galaxy S or Note series of top-end handsets, or whether it might launch a new range altogether.
The news follows reports that Apple is working on iPhones with bigger, curved displays which bend downward at the edges.
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