Okay, so this isn't the first time we've heard about a 7.85" iPad and a king-sized Kindle Fire. Now that the New York Times has jumped on the rumor bandwagon, though, the speculation seems that much more plausible.
Apple is indeed "developing a new tablet with a 7.85-inch screen," writes the New York Times. The device is probably going to cost "significantly less" than the $499 iPad and should launch later this year. The Times adds that it was tipped off by "several people with knowledge of the project who declined to be named discussing confidential plans."
Interestingly, the Times goes on to reveal that Apple's original tablet prototype, which dates back to the middle of the last decade, also had a 7" display. However, Steve Jobs reportedly "thought the device was too small and wondered aloud what it was good for 'besides surfing the Web in the bathroom.'" (The original iPad ended up with a 9.7" panel, and so has every other version of the device.)
Jobs also criticized 7" tablets in a conference call a couple of years back. He stated, "The reason we [won't] make a 7-inch tablet isn't because we don't want to hit that price point, it's because we think the screen is too small to express the software."
As the Times points out, though, Jobs was "famous for both 180-degree reversals of opinion and deliberate diversions intended to keep competitors away from a juicy opportunity," so it's conceivable he could have approved the 7.85" device—eventually. Then again, Jobs' instincts weren't always right. The original iMac was almost called the MacMan.
The Times has much less to say about the jumbo Kindle Fire. It only quotes a "developer briefed on Amazon's plans" as saying a new version of the Kindle with a larger screen is in the works. Older reports about that device suggest Amazon will roll out a second-gen version of the original Kindle Fire on July 31, so perhaps we'll hear about the bigger model around then.