The new prime minister will also reportedly ditch a ban on live animal exports in her first weeks in office.
The decisions will be a massive blow to campaigners who have spent decades lobbying for the reforms to spare animals from suffering.
Production of both fur and foie gras is considered so cruel that they are already banned in the UK.
All four measures were promised in the party’s animal-welfare action plan, announced last year to wide acclaim.
And curbs on live exports were promised in the Conservatives’ election manifesto, together with an end to hunting trophy imports.
But a senior Conservative told Politico: “Banning things seems very socialist. Informing people is the way to go.”
In February this year, right-wing cabinet members including Jacob Rees-Mogg intervened to block the Animals Abroad Bill, which contained the curbs on fur, foie gras, hunting trophies, and adverts for foreign theme parks that cause animal suffering.
The Kept Animals Bill, which banned live exports and keeping primates as pets and tackled puppy smuggling, could also be dropped. It had been due to be debated on Monday, which became the day of the Queen’s funeral, and no new date has been given.
A ban on cruel exports of live animals for slaughter and fattening had been hailed as a benefit of Brexit.
It would be a huge let-down, not only for those who work for these campaigns daily but also for millions of animals
The government says it is still looking at the fur and foie gras bans, but the source said the measures would not go ahead under Ms Truss, who appointed Mr Rees-Mogg as business secretary and promoted Mark Spencer, understood to have been another of those blocking the Animals Abroad Bill.
However, MP Scott Mann, who has spoken out in favour of a ban on live exports, has been promoted to environment minister.
Last week, Ms Truss sacked Zac Goldsmith as animal-welfare minister after he introduced reforms including an ivory sales ban and higher jail terms for cruelty. He also wanted to crack down on religious slaughter without stunning.
“A lot of his causes were very worthy, but you can be worthy when you’re the son of a billionaire,” the MP said in a bizarre comment. Lord Goldsmith’s father was financier James Goldsmith.
Lorraine Platt, co-founder of the Conservative Animal Welfare Foundation, told The Independent she was bitterly disappointed by news the bans would be dropped.
“It would be a huge let-down, not only for those who work for these campaigns daily but also for the millions of animals involved,” she said.
“Banning live exports and hunting trophies were manifesto commitments, and some people vote on manifesto commitments at elections.”
She said the foundation had often heard reports the measures could be scrapped or watered down.
Sir Roger Gale, patron of the foundation, condemned the “let them choose” argument as “a little spurious” and perverse when the UK has bans on producing fur and foie gras.
He told Times Radio he was concerned about the direction of travel of animal welfare under the new government, and millions of votes including in red-wall seats would be lost to the Tories if they reneged on animal welfare.
Foie gras production involves force-feeding ducks and geese with pipes pushed into their throats to fatten their livers.
Fur farms have been exposed as leaving animals suffering infected, bloody wounds, spreading disease and literally driving animals mad from confinement.
Claire Bass, executive director of Humane Society International/UK, said it was surprising and perplexing that senior Conservatives wanted to row back on the popular measures in last year’s animal welfare action plan.
“Animals matter to voters, and people will not be content with oft-recycled rhetoric about being a ‘world leader in animal welfare’ if it’s not accompanied by meaningful action,” she said.
“Banning fur imports is not un-Conservative, it’s simply the right thing to do in line with the British public’s moral compass.”
Under Boris Johnson, the government said it wanted to consider compulsory animal-welfare labelling on food and promised to consult on proposals next year.
The Department for Environment, Food and Rural Affairs (Defra) ran a 12-week consultation last year on new labelling standards for produce now that EU regulations no longer apply.
It also ran a consultation on banning fur imports, but has not released the results.
Ministers had at one stage said they would press ahead with the hunting trophy imports ban, but that pledge appears also to have been dropped.
Instead, they are backing a private members’ bill by backbencher Henry Smith that bans hunting trophy imports – body parts of wild animals killed by paying hunters. Mr Smith has called such hunting barbaric.
On foie gras, Defra said it was considering any further steps that could be taken, “building on the opportunities presented” by Brexit, and was still gathering information.
“The government has made clear that the production of force-fed foie gras raises serious welfare concerns,” a spokesman added.
On fur, the department said: “We are continuing to build our evidence base on the fur sector, which will be used to inform any future action on the fur trade.”
It also said the Kept Animals Bill would continue its passage through Parliament.
The Independent has also asked the office of the new Defra secretary, Ranil Jayawardena, to confirm whether the proposals will go ahead.