In late May, transmission operator Arqiva confirmed plans to shut down SeeSaw after failing to find an investment partner or buyer for the video on-demand service, which was launched in February 2010.
Criterion is understood to be on the verge of agreeing a last-minute deal to acquire SeeSaw, although no further details are known at this stage.
A source close to the talks told The Daily Telegraph: "Criterion are close to signing the deal to acquire SeeSaw. It's a good opportunity for them to access the lucrative online TV and film market, which Arqiva have been unable to crack."
An Arqiva spokesman said: "Arqiva is still in discussion with various parties about a possible sale of SeeSaw."
Criterion acquired Bebo last June for a reported price of around $10m (£6.17m), after AOL had acquired the social network only two years earlier for $850m. Former Channel 4 chief executive Michael Jackson now runs Bebo for Criterion.
Arqiva created SeeSaw after spending £8m to acquire the technology developed for Project Kangaroo, the BBC Worldwide, ITV and Channel 4 IPTV joint venture that was blocked by the Competition Commission.
SeeSaw was expected to officially shut down this week, but the website has been kept live while acquisition talks continue.