Blair also thinks an inquiry should be held into the leak to the Sun newspaper, his spokesman said, adding however that the decision should rest with Lord Brian Hutton, the judge who headed the probe.
Hutton was to issue his findings into Kelly's death at midday, but the Sun published what it called extracts of the conclusion which appeared to clear Blair of wrongdoing in the affair.
Kelly, a former UN arms inspector and respected expert on biological weapons, killed himself in July shortly after the Ministry of Defence, his employer, exposed him as the source of a BBC radio report in May which alleged that Downing Street had "sexed up" intelligence on Iraq in the run-up to war.
"The prime minister is very angry about the leak of Lord Hutton's report," his spokesman told journalists.
"The government and Alastair Campbell (Blair's former director of communications and a key figure in the Kelly affair) have categorically denied that they have had anything to do with the leak," he said.
"The Sun has said that the leak comes from someone with nothing to gain politically or financially from such a leak," the spokesman added.
"We believe there should be a leak inquiry involving all the parties, but this is a matter for Lord Hutton... Lord Hutton wishes to announce himself how he will proceed and we will co-operate with him fully."
Struggle for secrecy
Hutton went to great lengths to try to keep his report secret until its publication, giving only a few copies to key players - including Blair, Campbell, the BBC and Kelly's family - on written pledges that they would not divulge its content.
But in its Wednesday edition, the Sun claimed to have been given a precis of the judicial report, and said it criticised the BBC's editorial judgement rather than blaming the government for identifying Kelly.
Hutton is to make his findings
known on Wednesday at 12:30GMT
It quoted the author of the report, Lord Brian Hutton as saying "there was no dishonourable or underhand or duplicitous strategy by the government covertly to leak Dr Kelly's name to the media."
The Sun also said that Hutton would deliver only "a mild rebuke" to the Ministry of Defence for failing to warn Kelly in advance that his name would be confirmed to any reporter who guessed it.
In contrast, it quoted Hutton as concluding that an allegation by the BBC's defence correspondent Andrew Gilligan that the government "sexed up" its dossier on Iraqi weapons of mass destruction was "unfounded."
Hutton was to summarise his findings in a televised statement at the Royal Courts of Justice in London at 12:30 GMT, with Blair to make a statement and field questions in parliament from 2 pm.