Last week, private equity firms CVC Capital Partners, Kohlberg Kravis Roberts and The Blackstone Group admitted they were at the "preliminary stages of assessing Sainsbury's".
However, Cinven and Texas Pacific have also met to discuss a bid, according to reports in The Times.
“There have been preliminary discussions between Cinven and TPG since Saturday morning,” a source is reported to have told the paper.
However, analysts are doubtful any bid for Sainsbury's will materialise. “A mooted bid of 600p would be virtually impossible to put together, both in terms of financing, without a huge slug of equity, and a rate of return which we think would be well below that acceptable to VC investors,” said Seymour Pierce analyst Richard Ratner.
“We believe that, given the rise in the share price, the odds are against a bid appearing, and, even if it does, it is unlikely to be much above the current level,” he added.