Silicon Valley blog Valleywag cites an inside source at Apple, claiming that the firm will launch the bid in early June, in time for its Worldwide Developer Conference.
"A source who's plugged into the Valley's deal scene and has been recruited by Apple for a senior position says Apple and Twitter are in serious negotiations, with the goal of unveiling a deal by June 8, when Apple's annual Worldwide Developers Conference launches in San Jose," said Valleywag reporter Owen Thomas.
This is not the first time that Twitter has been the subject of buyout advances. In November the firm turned down a bid from Google which was said to be in the region of $500m.
Despite concerns raised by some over high churn rates and continued attacks against the service, the Valleywag report suggests that it will be the lure of Twitter's web savvy developers and users that appeals to Apple more than the actual technology, a version of which is available through the App Store.
"Users overshare every last detail of their lives, while Twitter makes these updates available on its web site, via RSS, and through third-party applications," wrote Thomas.
"Apple is surely realising it needs to play in this world, and needs someone to show it the way. Is it coincidence that Apple has put Twitter executives on stage so frequently, or that it profiled Twitter as a business recently?
"If Apple buys Twitter, it won't be about making money. It will be about making a statement. In 140 characters or less."
Apple recently posted a business profile of Twitter under the heading: Twitter. Triumph of humanity.
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