British spy chiefs have "genuine concern" about sources being exposed if classified parts of the wiretap request are made public, according to figures familiar with the discussions.
"It boils down to the exposure of people," said one US intelligence official, adding: "We don't want to reveal sources and methods."
Another said Britain feared it would set a dangerous "precedent" and make people less likely to share information, knowing one day it could become public.
The current row is deemed so politically sensitive that staff at the British Embassy in Washington DC have been barred from discussing it with journalists.
British Prime Minister Theresa May, who has a lukewarm relationship with Trump, has been kept at arms-length and is understood to have not raised the issue directly with him.
The row concerns an FBI request to wiretap Carter Page, a former Trump campaign foreign policy adviser, which was made in October 2016 - the month before the US election.
The FBI said it had suspicions that Page was being targeted for recruitment by the Russian Government and cited classified intelligence to make its case.
It was granted approval for 90 days of surveillance by a secret court established by the Foreign Intelligence Surveillance Act. It was renewed a number of times.
Trump wants to declassify 21 pages from one request. He announced the move in September, then backtracked, then this month was "very seriously" considering it.
Britain and Australia are understood to oppose the move.
Numerous spokesman for UK and US government bodies declined to comment.
Page has denied being a foreign agent for Russia.
