Tony Blair telephoned the chairman of the BBC's governors to reveal that a civil servant had admitted having contacts with Andrew Gilligan, the defence correspondent for the Today programme.
The Prime Minister told the Hutton inquiry he tried to persuade Gavyn Davies to retract Mr Gilligan's report that the Government had "sexed up" its dossier on Iraq's weapons of mass destruction.
Mr Blair called Mr Davies between 7am and 7.30am on Monday, 7 July, three hours before the Foreign Affairs Select Committee published the report of its investigation into the affair, "to see if we could find a way through this".
On the Tuesday, the Ministry of Defence issued a press statement that led to Dr David Kelly's name being made public.
The Prime Minister said he used the "entirely private conversation" to try to persuade the BBC to accept its broadcast was wrong. He told the inquiry: "I then said to him, and said, you know, in confidence actually over the last few days, we do have reason to believe there may be, we do not know at the moment, someone who has come forward as a source. It looks like from what he is going to say that he does not back up Mr Gilligan's story, but we cannot be sure of that."
Mr Blair said he told Mr Davies: "Whatever we think about your original broadcast and the allegations, is not the best way through for you to say, 'Well, we stand by our right to have broadcast the story but we accept the story was wrong', and we [10 Downing Street] say we accept that as a retraction and we can debate about whether it was right or wrong that the story should ever have been run, but none the less the BBC have now clearly retracted the original story?"
Mr Davies rejected Mr Blair's appeal for the BBC to issue a retraction. He told the inquiry: "At the very end of the conversation Mr Blair said he had become aware recently that someone had come forward to the MoD saying he was Mr Gilligan's source.
"I said, 'Bear in mind, Prime Minister, that Mr Gilligan has said he spoke to three or four people in preparing his reports, although he only had one principal source for part of them'. I said, 'I do not know who the source is, Prime Minister, but bear in mind he talked to three or four'."
Mr Davies said he had tried to avoid speaking to Mr Blair because of the row between the Government and the BBC. But he agreed to take a call from the Prime Minister that morning because the BBC's governors had met the previous night and agreed on a statement.
Mr Davies said: "Mr Blair said he [felt] ... the Gilligan story was wrong. He [thought] ...the BBC should retract the story. He thought perhaps a good way was for us to say it was valid to have broadcast it but we should now retract it."
Mr Davies said: "I explained ... that conspicuously the governors had not said Gilligan's source's allegations were intrinsically true and we had conspicuously said that we did not question the integrity of the Prime Minister himself. He said that was all very well and was grateful but still thought it should be retracted."