Sky News has learnt that Avant Homes, which is run by former Persimmon chief Jeff Fairburn, made an all-share proposal to the board of Crest last month.
The bid, which City sources said had been rejected by Crest's board in the last 10 days, would have seen Elliott become the biggest shareholder in the combined group, according to insiders.
Best of the Money blog
One said the combination risked representing an unattractive prospect for some Crest Nicholson shareholders.
All the information you need to see the results come in, wherever or however you like.
Avant's interest comes weeks after Crest rejected a bid from Bellway, another London-listed housebuilder, that would have valued the target at £665m.
Bellway now has a deadline set by the Takeover Panel to make a formal offer for Crest of next Thursday.
It is unclear if it intends to submit a revised offer.
"The board of Crest Nicholson evaluated the Revised Proposal with its financial advisers and concluded that it significantly undervalued Crest Nicholson and its future standalone prospects and was not in the best interests of Crest Nicholson's shareholders," it said last month.
"The board therefore unanimously rejected the revised proposal on 14 May 2024."
A prospective bidding war for one of Britain's largest housebuilders would thrust an industry that will be at the centre of the new Labour government's policy agenda back into the City spotlight.
Labour pledged in its manifesto to build 1.5 million homes during its first five-year term.
Read more on Sky News:Markets buoyed by Labour landslide after Tory turmoilLabour have won - but what happens next?Labour's journey back to power
Housebuilding stocks responded positively to Labour's landslide victory on Friday morning, with gains for Persimmon, Barratt Developments and Taylor Wimpey.
Please use Chrome browser for a more accessible video player
Persimmon is already contemplating a bid for Cala Group, which is owned by Legal & General, while Barratt and Redrow agreed a £2.5bn merger earlier this year.
Keep up with all the latest news from the UK and around the world by following Sky News
Be the first to get
Install the Sky News app for free
On Friday morning, shares in Crest Nicholson were up 1.4% to 248p, giving the company a market capitalisation of £628m.
