GameStop - once a stock market minnow that shot to fame during a meme-stock frenzy five years ago - is offering to pay $US125 ($A173) a share in a 50-50 mix of cash and stock, Cohen said in a letter to eBay's board.
Based on eBay's Friday close, the bid represents a premium of about 20 per cent.
EBay has a market capitalisation nearly four times larger than GameStop, making the buyout bid an ambitious attempt.
The US videogame retailer has already built up a five per cent stake in eBay through shares and derivatives, Cohen said in the letter, which was seen by Reuters.
Its unsolicited offer to buy the US online marketplace was first reported by the Wall Street Journal, citing an interview with chief executive Cohen, also GameStop's largest investor.
Cohen, who is pushing to boost the struggling videogame retailer's market value more than tenfold, told the Journal that putting eBay and GameStop under one roof would create huge opportunities to improve earnings and cut costs.
"It could be a legit competitor to Amazon," Cohen said about eBay to the Journal.
Cohen said in the letter that GameStop would cut $US2 billion ($A2.8 billion) of eBay's annualised costs within 12 months of close, resulting in an increase in the company's earnings per share.
GameStop's 1600 US locations would give eBay a national network for authentication, intake, fulfillment, and live commerce, he added.
He told the Wall Street Journal he was prepared to pursue a proxy fight if eBay's board was not receptive to the proposal.
EBay did not immediately respond to Reuters requests for comment on GameStop's offer.
"EBay should be worth - and will be worth - a lot more money," Cohen said in the interview. "I'm thinking about turning eBay into something worth hundreds of billions of dollars."
Cohen, dubbed the "meme king" by retail traders for his role in the 2021 meme-stock frenzy and his outsized influence among individual investors on social media, has built a reputation for bold, unconventional bets that can move markets.
Cohen said he has already lined up financial commitments, including a commitment letter for about $US20 billion ($A28 billion) in debt from TD Bank, and may seek backing from external investors including Middle Eastern sovereign wealth funds for the deal, according to the WSJ report.
Cohen said that following the close, he would serve as the chief executive of the combined company.
GameStop had a market value of nearly $US12 billion ($A17 billion) at the close of business on Friday, while eBay had a market value of about $US46 billion ($A64 billion).
Their shares have gained 32.1 per cent and 19.5 per cent, respectively, in 2026.
