Twitch, which enables gamers to upload live streams of themselves playing games, claims 45 million visitors a month and would give any acquiring company a large presence in the gaming world.
The paper said Twitch may ultimately choose to raise funding from investors rather than sell itself to Google, Microsoft or any of the other gaming companies that have expressed interest.
But if the YouTube deal goes ahead, Google is reportedly expecting a battle with regulators, which might argue that the deal stifles competition with other internet video providers.
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Talks with Google are expected to centre on how independent the Twitch brand will remain, according to an anonymous source cited by The Verge.
Twitch is said to believe that Google can help it become the definitive platform for watching and streaming live video gaming.
The company raised $20m from investors in 2013, but some market commentators believe capital alone will not enable Twitch to scale its technology and infrastructure to keep pace with its growth.
Twitch reportedly believes that Google may be the right partner to help it handle huge amounts of live and user-generated video on a global scale.
And although YouTube is the largest online video platform, its live-streaming services are not nearly as widely adopted as Twitch's, notes the Business Insider.
The proposed deal has been compared to Facebook's $1bn acquisition of Instagram in 2012, in which an established firm acquired a fast-growing startup that was a potential competitive threat.