Facebook, based in Menlo Park, Calif., has been buying a number of companies in recent months to increase its presence in the world of smartphones.
The Opera browser runs on desktop computers, various phones and tablets and even the Nintendo Wii, and the company claims to have more than 200 million users, the Los Angeles Times reported.
As of last week, Google's Chrome browser was the most-used Web browser, and was thought to be the main target for Facebook.
"The move -- which would no doubt send shivers of panic through Google -- although unlikely to affect Chrome's continued growth in the short term, would see the two tech giants battle it out on your desktop and mobile for Web surfing as well as social networking," Pocket-lint said.
A report by The Next Web quotes sources close to Opera as saying the company has been looking for potential buyers.
A spokeswoman for Facebook said the company did not wish to comment on the rumored purchase, while a spokesman for Opera said the company does not respond to market rumors, the Times reported.