David Cameron asked cabinet secretary Sir Jeremy Heywood, the country's most senior civil servant, to contact The Guardian about the classified material handed over by Edward Snowden, it has been reported.
The intention was to spell out the serious consequences of continuing to publish material about UK and US intelligence operations, The Independent said.
The Guardian editor Alan Rusbridger revealed his newspaper had destroyed a hard drive containing a copy of the secret documents under the supervision of GCHQ officers following sustained pressure from government.
On Tuesday the United States appeared distanced itself from the prime minister's decision to ask his officials to put pressure on the newspaper to destroy its files.
Asked by reporters whether the Obama administration would take similar action in the US, White House deputy press secretary Josh Earnest said: "It's very difficult to imagine a scenario in which that would be appropriate."
The fresh claims about Number 10's role followed earlier confirmation by home secretary Theresa May that she had been briefed in advance about the possible detention of David Miranda, partner of Guardian journalist Glenn Greenwald, at Heathrow Airport.
On Monday Earnest said the British government gave the United States advance notice that London police intended to detain Miranda but added that the US did not request the detention and was not involved in the decision.
Greenwald - the reporter who interviewed American whistle-blower Edward Snowden - called Miranda's detention a "profound attack on press freedoms and the news-gathering process".
May revealed yesterday that she had been briefed in advance by police that they may detain Miranda.
"I as home secretary do not tell the police who they should or should not stop at ports or who they should or should not arrest. I think it's absolutely right that that is the case, that the police decide who they should stop or not and whether they should arrest somebody or not," she said.
A Downing Street source told the Press Association on Tuesday that David Cameron was "kept abreast of the operation in the usual way" but denied any political involvement in the decision.
Scotland Yard has defended the detention as "legally and procedurally sound". But several politicians and civil liberties campaigners have raised concerns about how the use of anti-terror laws to detain and hold someone connected to a journalist.
Lib Dem MP Julian Huppert, a member of the influential Commons home affairs committee, told The Huffington Post UK that the incident "raises questions about what the home secretary's and prime minister's involvement was".
"I am very alarmed about it, it does seem to be an abuse," he said. "Hassling people around journalists, that is clearly nothing we want to see."
Miranda said he was questioned by six agents on his "entire life" while being held at Heathrow for nine hours - the legal limit before a suspect must be charged or released.
He was stopped at 8.30am on Sunday when returning from a trip to Berlin. He was questioned under schedule 7 of the Terrorism Act 2000 which applies only at airports, ports and border areas, allowing officers to stop, search, question and detain individuals.
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