There are "no excuses" for the release of hundreds of foreign prisoners at the end of their sentences without being deported, the prime minister has said.
Mr Blair turned down Mr Clarke's offer of resignation
Mr Blair was quoted by the News of the World as saying Charles Clarke's future as home secretary "depends on what happens" with those released.
But Downing Street later insisted he remained "supportive" of Mr Clarke.
Mr Clarke faces increased pressure to quit after revealing five immigrants released from prison had re-offended.
A Downing Street spokesman said the prime minister was "totally supportive of the home secretary in resolving the problems that he has been addressing over the last few days".
Hunt for offenders
Police officers are undertaking raids to detain some higher risk offenders.
The Home Office have not revealed how many have been tracked down so far by the 200 officers involved in the hunt.
"Operations to detain the most serious offenders are ongoing, and we now have more individuals in detention pending deportation and removal from the country," a Home Office spokeswoman said.
"Work will continue through the weekend," she added.
In the News of the World interview, Mr Blair said he had not accepted Mr Clarke's offer to resign because his home secretary had made efforts to rectify the situation.
But he said he was "not going to speculate" on whether Mr Clarke would be out of a job if a serious crime was committed by one of those released.
"It depends on what happens, what the reasons are," he added.
Transport Secretary Alistair Darling has defended Mr Clarke, saying he still has the authority to carry on as home secretary.
"He is a good minister. When something like this goes wrong it's not surprising as he did he thought, 'Well should I consider my position, should I go?'" he told Andrew Marr on BBC One's Sunday AM programme.
"I think the view was that he should carry on, he should sort this out."
Mr Darling said the primary concern was to make sure this did not happen again, adding he hoped Mr Clarke would still be home secretary in a week's time.
But he admitted that "last week was bad, my prediction for this week is that it will be bad as well".
The prime minister's interview comes after the home secretary said that five of those released had re-offended and had been convicted for offences relating to drugs, violent disorder and inflicting bodily harm.
Two have also faced rape claims, with one case dropped because of lack of evidence.
Mr Blair said he had been "pretty angry" when he was first told about the releases.
"But it was important to get to the facts and see what people had been trying to do," he said.
"I make no excuses for what happened, it was wrong and shouldn't have happened.
1,023 freed without being considered for deportation
79 of them originally jailed for more serious crimes, including 13 of them for murder, manslaughter, rape or child sex offences
Five of them known to have committed offences since release - including violent disorder, drug crimes and grievous and actual bodily harm
Deportation action now started for 63 of the 79 more serious criminals
Officials have decided nine of the offenders should not be deported
"The reason the problem was uncovered was because people started to make changes. We had no knowledge until we started to work through the system about whether foreign prisoners were being deported."
There was "no excuse" for the government not to have sorted the problem but it was "a systematic failure that goes back years", Mr Blair added.
It was fair to ask at what point a politician became responsible, Mr Blair said.
"But he has been acting on it," he added.
"That's where I disagree with people who say he's ignored it - he hasn't."
On Saturday, the Home Office said more than 200 police and immigration officers were taking part in raids across the UK, to track down 63 high-risk foreign offenders, following initial raids overnight on Friday night.
They are among the 1,023 prisoners released into the community without being considered for deportation over the past seven years - 79 of whom had been in jail for serious offences.
Mr Clarke is continuing to face calls from opposition MPs for him to quit.
Conservative leader David Cameron said the home secretary "had presided over a system that had manifestly failed" and it was time to step down.
Shadow home secretary David Davis said Mr Clarke was a "national tragedy".
"This has actually done the opposite of what his proper job is, and that is to protect the safety of the public," Mr Davis told Sky News TV.
"If I were in his position, I would tender my resignation to the prime minister, and I wouldn't take no for an answer," he added.