No one was available to comment on whether the fund planned to hold onto its 25 per cent stake in Sainsbury, but it said in a statement that it remained supportive of the retailer’s operational strategy.
"It doesn't look like they're sellers. But it's hard to say what Tchenguiz's plans will be," Nick Bubb, an analyst at Pali International, said, referring to the 10 per cent stake Robert Tchenguiz, a property entrepreneur, holds with the supermarket chain.
"Since Delta Two's original proposal was submitted to the board of Sainsbury, the required funding and cost of capital has increased significantly, which has adversely affected the investment case"
Delta Two statement
Delta Two's withdrawal marks the second failed bid for Sainsbury in less than seven months.
CVC Capital Partners, a private equity firm, abandoned a takeover attempt in April, following opposition from the company's board and the Sainsbury family - which owns about 18 per cent of the supermarket group.
At 0830 GMT Sainsbury's shares were down 17.5 per cent at 457.75 pence on Monday - cutting a large slice out of Delta Two's stake in Sainsbury.
The fund bought most of its stake in April, at a price which market sources said was around 575 pence a share. It later bought the rest of its stake at around 595 pence a share.
Shares in Wm Morrison Supermarkets, which has also been tipped as a possible bid target, fell as much as 7 per cent.
Both Sainsbury and Morrison have attracted speculation about possible bids in part because of their rich property assets.
The Sainsbury board opened its books to Delta Two after it proposed a 600 pence-a-share cash offer, which was above CVC's proposal of 582 pence.
However, the Sainsbury family remained reluctant to sell out at that price.
Delta Two said last month it needed an extra 500 million pounds of funding for its bid from its backer, the Qatar Investment Authority (QIA).
An anonymous source said on Sunday there was no certainty the QIA would provide the extra money.
"Since Delta Two's original proposal was submitted to the board of Sainsbury, the required funding and cost of capital has increased significantly, which has adversely affected the investment case," the Qatari fund said in a statement.