The Conservatives are calling for correspondence that raised concerns about the recording of crimes committed by Britons overseas to be published.
Mr Reid said the matter was a "very serious problem"
They want a letter the Association of Chief Police Officers sent to minister Tony McNulty in October - and a reply by minister Joan Ryan - made public.
It warned the police were having problems tracing hundreds of dangerous British criminals convicted overseas.
Home Secretary John Reid says the letter may be printed after an inquiry.
The home secretary promised a full inquiry after the Home Office was accused of ignoring files on criminals convicted overseas.
Downing Street said the prime minister has full confidence in Mrs Ryan, who is under pressure to explain her role in the controversy.
The prime minister's official spokesman said there was no evidence ministers knew of a backlog of cases or the scale of the backlog.
Out of 27,500, police say 540 were serious offenders - including rapists, paedophiles and murderers. So far, just 260 have been identified.
There is concern offenders could have been cleared to work with children.
Initially Mrs Ryan, the minister responsible for the Criminal Records Bureau, said ministers had not been aware of the issue until this week.
But the Association of Chief Police Officers (Acpo) revealed it had written to police minister Mr McNulty in October.
Acpo said it had indicated in the letter to Mr McNulty that there were "continuing difficulties around the exchange of criminal records across the UK".
The difficulties were "specifically the need for notifications from EU states to be accompanied by biometric data", it said in a statement.
The letter "did not relate to funding issues nor the outstanding notifications", the statement added.
Acpo says that, at the end of September, it also made a request to a Home Office official for more money to help deal with the backlog of 27,500 cases.
The Home Office said Ms Ryan had dealt with the first letter, while the request for more funds was dealt with by an official, not a minister.
The department still insists no minister was aware of the backlog in cases until this week.
Shadow home secretary David Davis said the position of the two ministers would be "untenable" if they knew of the problems with the offenders' details.
"We need to see the full copy of this letter. Acpo should now publish it including all replies," he said.
BBC political editor Nick Robinson said Acpo was saying the letter had highlighted difficulties with the procedure, while Mr Reid was saying the letter had largely praised the new system.
Our political editor said Mr Reid described the opposition's attempt to get the letter published as "mischief-making".
'Catalogue of blunders'
In March 2006 Acpo took over responsibility for logging the cases with the police database.
On Tuesday, Acpo's Paul Kernaghan told a Commons committee that details of 27,529 cases, including 25 Britons convicted of rape, had been "left in box files" at the Home Office before his team had taken over.
The cases involved included:
25 rapes3 attempted rapes29 paedophiles17 other sex offenders5 murders9 attempted murders13 manslaughter convictions29 robberies
He said Acpo was now working through the backlog.
Mr Reid made a statement to the Commons on Wednesday saying that the details of 260 of the 540 serious criminals had now been entered into the PNC.
"The remaining 280 cannot be entered on the computer and are the subject of further inquiries to the notifying country to get more details to try to establish the identity of the offender," he added.
'Worst three years'
An inquiry into the Home Office's handling of the details was expected to be completed within six weeks, Mr Reid said.
Mr Davis attacked the department's recent record as a "catalogue of blunders".
"The last three years have been the worst three years in the Home Office's 200-year history," he said.
The Liberal Democrats also criticised the government, saying the mixture of "ignorance and incompetence would be comic if it did not jeopardise public safety in the way that it has".
Earlier, Tony Blair and Conservative leader David Cameron had clashed over the issue in a heated Prime Minister's Questions.