The Qatari-backed fund is understood to be planning to pump an additional £500m cash into its offer following concerns from the Sainsbury's family that there is too much debt in the deal.
The increase could come as early as this week, according to reports in The Times.
A source also dismissed speculation that Sainsbury's had demanded Delta Two to make its best and final offer or walk away. “There's progress being made and I don't think they're in that kind of position yet,” the source told the paper. “The mood is still friendly.”
Meanwhile, reports suggests the drawn out talks between the two sides could be nearing a conclusion.
Reports in the Guardian suggest an indicative offer could be tabled by Delta Two this week, while the Daily Telegraph reports that Sainsbury's wants talks to end within three weeks.