Private equity firm Silver Lake Partners bid as low as $11.22 per share for Dell Inc last summer, when it first discussed a buyout with founder and CEO Michael Dell, according to a person familiar with the situation.

Since then, Silver Lake and Dell have raised their bid to take the world's No. 3 personal computer maker private to $13.65 a share. Major shareholders including Southeastern Asset Management have protested that the company remains severely undervalued at the current bid.

Separately, Southeastern, the largest independent shareholder of Dell with a 8.4 percent stake, broached the possibility of a leveraged buyout to Michael Dell and expressed interest in contributing its equity toward any deal, a second person familiar with the matter said.

Both sources asked not to be named because the matter is not public. Dell did not immediately respond to requests for comment.

A clearer picture of the negotiations leading up to the deal is expected to emerge in the last week of March in a company proxy filing.

Silver Lake gradually raised its offer price to the current level, one of the two sources said.

CNBC first reported the opening bid. According to the business network, private equity house KKR & Co LP had also discussed a bid for Dell at $12 to $13 a share but dropped that offer in December.

Investors, including No. 3 shareholder T. Rowe Price, have publicly objected to the proposed $24.4 billion buyout offer led by the company's founder and Silver Lake.

Michael Dell is trying to complete his company's transition from a low-margin PC maker into a provider of computing services. The makeover has become more urgent as the PC market shrinks. Analysts say it might best be carried out if the company were taken private, away from public shareholder pressure and scrutiny.

(Reporting By Greg Roumeliotis and Soyoung Kim in New York)