The Wall Street Journal reported Tuesday that Apple has asked its suppliers to produce "between 70 million and 80 million units combined" of the devices, with 4.7-inch and 5.5-inch screens, by Dec. 30. The paper cited "people familiar with the matter."
The report is consistent with images and videos of the "iPhone 6" leaked in recent months on technology websites that suggested Apple is making new models with bigger screens than the 4-inch iPhone 5, which was released in 2012. That model was replaced with the iPhone 5C and iPhone 5S in 2013.
Sites such as MacRumours, TechRadar and AppleInsider say they expect the new smartphones to be released in September, although there are some who suggest the larger, "phablet" model may be released later.
Some of Apple's competitors already make larger screen models.
One of them, Samsung, released a new ad this week, poking fun at Apple's rumoured larger-screen iPhones.
The ad for the Galaxy S5, which has a 5.1-inch screen, is titled "screen envy," and features a man with an iPhone asking a friend with a Galaxy S5 if he's heard a rumour that iPhone might be getting a bigger screen.
A voiceover goes on to suggest that iPhone users have been waiting "for like two years," and that, in fact, the Galaxy S5 is "everything you've been waiting for."