The home secretary says he told Tony Blair he was ready to quit over the 1,023 foreign prisoners released but not considered for removal from the UK.
Figures show 288 of the prisoners were freed after officials became aware of the problem last summer.
Mr Clarke, who will brief MPs at 1230 GMT, said he told the premier of the situation just before Christmas.
Shadow home secretary David Davis said Mr Clarke's position was becoming "increasingly untenable".
The released prisoners, which included three murderers, nine rapists and five child sex offenders, should have been considered for deportation under the Home Office's own rules. The problem stretches back to 1999.
Of the 1,023, 160 were recommended for deportation by their trial judges.
Mr Davis said it should not have been difficult - as Mr Clarke has claimed - to deport these prisoners.
"How hard can it be to organise to take prisoners from the prison door to the airport or the port?," he told BBC Breakfast.
Mr Clarke blamed the bungle on a breakdown in communication between the Immigration and Nationality Directive (IND) and the Prison Service.
The prime minister's official spokesman said "every effort" will be made to trace released foreign prisoners by the end of April.
He also said ministers had only become aware in February that an extra 288 foreign nationals had been released since last August, despite the changes demanded by ministers.
Of those 70 were now "being considered" by officials and 14 had already been deported.
Asked why the Home Office had not tackled the problem when it first came to light, Mr Clarke said it was "like the old story of the tanker turning around".
He was also asked if he had told the prime minister he was ready to quit over the debacle. Mr Clarke said: "Yes I did. I told him I was prepared to resign if he thought it was right. He said he didn't think it was right."
So far the IND has located 107 of the total, leading to 20 deportations.
7 convicted of other sex offences
57 convicted for violent offences
2 guilty of manslaughter
20 drug importers
54 convicted of assault
27 convicted of indecent assault
As well as the three that had been convicted of murder, the nine of rape, and the five of sex offences against children, a further seven had served time for other sex offences, 57 for violent offences and two for manslaughter.
There were also 41 burglars, 20 drug importers, 54 convicted of assault and 27 convicted of indecent assault among those freed.
The Home Office said it did not have full details of offences committed by more than 100 of the criminals, but 237 were failed asylum seekers and 54 were still having their asylum applications considered.
Liberal Democrat leader Sir Menzies Campbell said it was "extraordinary" that so many people convicted of serious offences had "simply disappeared".