Silver Lake consortium, which includes Microsoft, has offered Yahoo $16.60 per share for a minority stake, while TPG Capital has offered a dollar more.
The offers for a 19.9% stake value Yahoo at around $20.6bn, which is less than half the amount offered by Microsoft in 2008.
The Yahoo board is keeping the sell-off to under 20% of its equity to avoid a shareholder vote on the issue, and is reportedly considering the bids, although they are lower than what recent Yahoo investors were expecting, according to US reports.
Analysts say that investors who have bought Yahoo recently were hoping for a significant premium and a take-out of the full company.
Chinese internet trading site, Alibaba, in which Yahoo has a 40% stake, has indicated an interest in buying the whole company.
Yahoo's board must now consider whether to sell a stake or invite offers for the company as a whole.
Pundits say if either Microsoft or Alibaba acquire part or all of Yahoo, it could shake up the internet search market, particularly as the Silver Lake consortium includes Marc Andreessen, co-founder of Netscape, who could become Yahoo’s new CEO.
Microsoft is keen to acquire a minority shareholding to protect the 10-year agreement with Yahoo, announced in 2009, according to The Guardian.
The two firms share search technology and advertising revenues in an alliance designed to boost Microsoft's Bing search engine against dominant search rival Google.
Yahoo's share price has fallen by more than 7% this year and 44% in the past five years in the face of stiff competition from rivals like Google and Facebook.
Ex-CEO Carol Bartz was ousted in September after her plans to turn the company around failed to deliver results after nearly three years at the helm.
After firing Bartz, Yahoo's board set up an executive leadership council made up of key executives to conduct a comprehensive strategic review to position the company for future growth and appointed financial advisors to find potential buyers.
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