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Billionaire Dan Loeb, who for months has been agitating for a sale of Campbell Soup, has privately begun to admit to business associates that such a deal isn’t likely in the near term, sources told The Post.

The revelation is yet another setback in the activist investor’s bid to make a bundle off his investment in the troubled Camden, NJ-based soupmaker.

On Wednesday, Campbell announced that four descendants of John Dorrance — the inventor of condensed soup and early owner of Campbell — intend to back the current board at next month’s shareholder meeting. The Dorrance contingent controls more than 40 percent of Campbell’s shares.

The announcement was a double-whammy for Loeb: While he needs a simple majority of shareholders to win the 12 board seats he seeks, he would need the support of two-thirds of shareholders to get a sale approved.

That now looks nearly impossible with the Dorrance shares now pledged against him.

Earlier this month, The Post reported that Kraft Heinz looked at Campbell but passed amid doubts about long-term demand for canned and packaged foods as shoppers become increasingly health conscious.

In an Oct. 1 presentation to shareholders, Loeb’s hedge fund Third Point appeared to signal uncertainty about a sale, as it listed such a deal as one of “multiple paths to create shareholder value.”

But with roughly 40 percent of votes claimed by the Dorrance group — some of whom Loeb hoped to convert — Loeb’s chances of getting on the board are beginning to look downright remote.

The Dorrance list includes Campbell board member Archbold van Beuren, a trustee of a Campbell voting trust that owns a 5.9 percent stake.

“The fact that Loeb said the existing board was horrific made it hard for him to side with Loeb,” a source who knows van Beuren said.

Shareholders more deeply dug in against Loeb include board members Bennett Dorrance, with a 15.1 percent stake and Mary Alice Dorrance Malone, with 17.7 percent.

Dorrance descendant Charlotte C. Weber, who is not a board member but was considered a “swing vote,” also pledged her shares, bringing the Dorrance total to nearly 41 percent.

“Dan probably should have reached out to her more,” a source said of Weber.

Despite the setbacks, Loeb’s camp kept up the heat, firing back at Campbell Wednesday morning.

“It is hardly news that the entrenched family owners, who have long enriched themselves at the expense of shareholders and the company, seek to preserve their board seats and reign of error,” Loeb said in a statement.

“This group of billionaire heirs and heiresses are attempting to intimidate smaller shareholders by flaunting their inherited voting bloc as an impenetrable moat,” he added.

Campbell’s announcement comes weeks before shareholder proxy service ISS makes a recommendation on whether to vote for Campbell’s nominees or Loeb’s.

Proxy recommendations are often — but not always — followed by institutional shareholders like BlackRock and Vanguard, who combined hold 11.6 percent of shares.

Loeb may also be holding out hope that mom-and-pop investors side with the him.

Nearly 15 percent of Campbell’s shares are thought to be held by retail investors, a source estimated.

Loeb’s Third Point formally launched its campaign against Campbell in August but began building its 7 percent stake in the company weeks after the sudden departure of Chief Executive Denise Morrison in May.

Loeb is joined by Dorrance descendant George Strawbridge, who has a 2.8 percent stake in the company.

Campbell shares fell 5.4 percent, to $36.32 on Wednesday.

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