The home secretary must update the House of Commons on the hunt for 1,023 foreign criminals released without being deported, the Conservatives say.
The Lib Dems say Mr Clarke has lost the public's confidence
Shadow home secretary David Davis said he wanted an "urgent statement" on how many had been arrested and sent home.
Charles Clarke has faced calls to quit, but told a local paper in his Norwich constituency that he hoped to stay on.
It emerged he took three weeks to tell the prime minister the criminals, including murderers, had been freed.
The Home Office said Tony Blair was briefed when officials were in a position to give him full details.
The mistakes occurred over the past seven years, but 288 immigrants were released after Mr Clarke knew of the problem.
'Refusal to answer'
Mr Clarke said in a statement on Friday that 79 of the freed prisoners were serious offenders, of which 63 were now being deported.
But Mr Davis told the BBC on Monday the Home Office had refused to tell him how many of the 1,023 had been tracked down, and how many of the 63 have been arrested.
He said: "It's not good enough for Charles to say some time this week, perhaps after the local elections, I'll come back and tell you.
"I want to see an urgent statement in the House of Commons tomorrow so the public can know that these people who are putting them at risk are being brought in."
Conservative leader David Cameron accused Mr Clarke of "great incompetence".
"Charles Clarke's problem is that in July last year he was told of serious problems and he failed to take the right action," Mr Cameron said.
Mr Clarke has admitted a National Audit Office report in July 2005 first alerted him to a possible problem of foreign prisoners being released because deportation was not being arranged.
Mr Cameron said: "This incompetence goes to a deeper problem within government. This is a short-term government that doesn't look at the long term."
He also accused New Labour of an "obsession with headlines instead of sound administration", adding: "A government that lives by headline will die by headline, and deservedly so."
The Liberal Democrats have called for the home secretary to resign, saying he no longer commanded the public's confidence.
Mr Clarke, who is MP for Norwich South, admitted in an interview with the Eastern Daily Press he would only keep his job if he could maintain support of his colleagues.
"My own political future depends on my own strength of character dealing with the points that are raised," he told the paper.
"Secondly on the support of my own political colleagues, and thirdly and most importantly, on the decisions of the prime minister.
"If I lost that support, that would be different. I hope I will continue as home secretary."
But he said the pressure he was under "doesn't remotely compare" to the pressure of dealing with the 7 July London bombings.
Labour backbencher Shahid Malik said the prime minister had given 100% backing to Mr Clarke, who now needed more time to put the matter right.
Series of raids
On Friday Mr Clarke said five of the freed immigrants had re-offended and been convicted for drugs offences, violent disorder and inflicting bodily harm.
Two have also faced rape claims, with one case dropped because of lack of evidence.
The Home Office has not revealed how many of the released prisoners have been tracked down so far by the 200 officers involved in the hunt.
Over the weekend police launched a series of raids on addresses throughout the UK.