Eddy Cue, who oversees critical Apple businesses such as the iTunes store, Apple Music and iCloud, broached the idea of a bid at a meeting at the end of last year with Olaf Olafsson, Time Warner's head of corporate strategy, the people said.
The meeting at the media group's Manhattan headquarters had been arranged to discuss other commercial relationships between the two companies, such as the potential inclusion of Time Warner's cable channels in a future Apple video streaming service.
Discussions about Apple buying Time Warner did not get beyond a preliminary stage and never included Tim Cook, Apple's chief executive, nor his Time Warner counterpart, Jeff Bewkes, people familiar with the matter said.
Apple and Time Warner declined to comment.
The fact that Apple considered bidding for one of the world's most prominent media companies — Time Warner has a market capitalization of almost $60bn — underlines the tech group's growing desire to offer its own content.
It also points to Apple's willingness to consider new areas of growth as sales of its key device, the iPhone, enters a phase of slower growth and its cash pile mounts to $216bn.
The company, which earlier this month invested $1bn in Didi Chuxing, Uber's main competitor in the China, has already begun to produce content. It recently commissioned a video series about the app economy and a scripted series starring and produced by Dr Dre, the hip hop star, for the Apple Music streaming service.
However, these are small steps compared with peers such as Amazon and Netflix, which are spending billions of dollars a year on original series and movies.
Apple intends to ramp up its spending on original content to "several hundred million dollars a year", according to people familiar with the matter. It has also not ruled out acquiring a media company, a person close to Apple told the Financial Times.
Time Warner would be an obvious fit. The company and Walt Disney are the only leading media players that do not have dual share structures and controlling family shareholders. Comcast, 21st Century Fox, CBS and Viacom are all controlled by founders and their families, potentially making it harder to launch a takeover.
Time Warner also has highly sought-after assets: HBO, which produces Game of Thrones, Silicon Valley and Veep; Warner Bros, Hollywood's largest producer of films and television shows; and Turner, which owns several cable channels and holds the rights to NBA basketball.
The company received an unsolicited offer from Rupert Murdoch's 21st Century Fox just two years ago. On the day of the bid, the cash and stock offer valued it at about $86 a share, but Time Warner rejected it. Time Warner shares closed at $73.36 on Wednesday.
It is not clear Apple will revive its interest in Time Warner but bankers say the tech company has recently been considering a range of potential media targets. "They've been on the lookout for content assets for several months," said one that has worked with Apple.
Several bankers said Apple was more likely to go after a streaming company such as Netflix than a pure content player, as it would make it easier for Apple's services to continue to offer a wide range of content makers.
(Disclosure: Comcast is parent of NBCUniversal and CNBC.)