The planned legislation underpinning the deal to cede the islands to Mauritius, which needs Washington's backing, would not be included in the government's next parliamentary agenda, The Times newspaper said on Saturday.
Prime Minister Keir Starmer's office said London would try to persuade Washington to give its formal approval.
Trump said in February the deal was a "big mistake", having previously said it was the best Starmer would get.
Under the deal, Britain would retain control of the strategically important military base on Diego Garcia on a 99-year lease that would preserve US operations there.
A British government spokesperson said ensuring the long-term operational security of Diego Garcia would remain a priority.
"We continue to believe the agreement is the best way to protect the long-term future of the base, but we have always said we would only proceed with the deal if it has US support," the spokesperson said.
"We are continuing to engage with the US and Mauritius."
Britain forcibly displaced up to 2000 indigenous Chagossians in the late 1960s and 1970s to establish a base on the Diego Garcia atoll.
Toby Noskwith, a spokesperson for the campaign group Indigenous Chagossian People, said there had been some hesitation about the deal from the start among senior people in the Trump administration, perhaps even the president himself.
"We are astonished to have come to this point," Noskwith said.
"This has been framed mainly as a state-to-state issue but the people who have been lost throughout the process are the Chagossians, particularly elders and survivors."
He said questions needed to be asked about "the enormous sums of money which have been wasted on a collapsed negotiation, and the legality of conceiving a plan which denied the Chagossians their right to self-determination".
He said Starmer must facilitate the dignified resettlement of the Chagossian people.
The alliance between Washington and London has come under strain in recent weeks over Starmer's reluctance to get involved in the US-Israeli war on Iran and his refusal at the start of the conflict to allow Trump to use British air bases to launch attacks.
US forces have since been permitted to carry out what the prime minister calls defensive strikes.
Trump has also repeatedly criticised the British leader, saying he was "not Winston Churchill" and had ruined what is often called a "special relationship" between Britain and the US.
