Beverley Hughes was forced to resign over fraudulent visa issues
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A ban on visas to people from Bulgaria and Romania is to be partially lifted.
The suspension was introduced amid fears of visa fraud and the ban will remain in place for managed migration schemes for the self-employed.
But the Home Office said "those categories ... which are not in question" could be reintroduced.
Immigration minister Beverley Hughes quit the government last week over the issue of Romanians and Bulgarians using forged documents to enter the UK.
Emergency summit
The Tories say an independent inquiry into the UK's immigration system is needed following a series of allegations about the way it is run.
Speaking on Tuesday, after Tony Blair's emergency Downing Street summit on the issue, Mr Davis said an inquiry into asylum statistics announced by Mr Blair would not be able to investigate ministerial actions or policy.
"The system is not working...it's a complete shambles," he told the BBC.
'Rehashed' initiatives?
Home Secretary David Blunkett, who attended Tuesday's summit along with cabinet colleagues and security chiefs, said "the world is going a little bit
mad", as he tried to draw a line under the immigration issue.
Mr Blair unveiled a National Audit Office (NAO) probe into asylum statistics to restore public faith.
Mr Blunkett announced fresh crackdowns on bogus students and sham marriages.
But Mr Davis said these measures were a "rehash" of failed former initiatives and added the results of the summit were "frankly, fairly complacent".
Mr Blunkett conceded there was nothing new in his pronouncements but insisted
the event would produce results in the same way as a previous summit on street crime.
"It supported me in turning the problem round," he said.
'Sensible'
However, Mr Davis told BBC Radio 4's Today programme he had asked for the NAO to look at asylum statistics a month ago.
"The NAO cannot investigate ministerial actions, it can't investigate policy," he said.
Bogus marriages were not the only problems, there were also bogus companies and bogus colleges, he added.
"We know from internal documents that ministers have actually discouraged people from pursuing these issues."
Mark Oaten, the Lib Dems home affairs spokesman, said bringing the NAO in to review asylum figures was "a sensible measure", but he was disappointed there would be no independent inquiry.