Failure to deport the equivalent of a "small town" of foreign criminals will lead the public to "question the point" of Britain remaining in the European Union, according to a powerful cross-party committee.
MPs said it was "deeply concerning" that 5,789 offenders from overseas were walking the UK's streets - the highest number since 2012.
But Home Secretary Theresa May hit back, saying the Government removed a "record number" of foreign national offenders last year.
The Home Affairs Committee said the Government must take urgent action to "significantly" reduce the 13,000 total number of foreign convicts in the country.
In a damning report, it found the top three countries the criminals hailed from were all within the EU, with Poles accounting for nearly one in 10 foreign offenders - some 983 - while 764 were from Ireland and 635 from Romania.
MPs criticised the Home Office for consistently failing to remove convicts despite their home nations being part of the 28-member bloc and warned the problems undermine public faith in Britain's EU membership.
"The clear inefficiencies demonstrated by this process will lead the public to question the point of the UK remaining a member of the EU," the report said.
The committee criticised Mrs May for being "unconvincing in suggesting that remaining a member of the EU will make it easier to remove these individuals from the UK".
It also warned it would take a "modern miracle" for David Cameron to meet his "no ifs, no buts" pledge to cut migration to under 100,000. Latest official figures showed net immigration was 323,000 to September 2015, an increase of 31,000 on the previous year.
And MPs said it was "deeply concerning" that there had been so little improvement in the immigration backlog, which stood at around 345,400 at the end of last year.
In 2015-16, 5,692 foreign criminals were removed from the UK - the highest since the series began in 2009.
Of these, 3,451 were from the European Economic Area, compared to 1,019 in 2010/11.
Mrs May said: "Foreign nationals who commit crimes here should be in no doubt of our determination to deport them.
"Last year we removed a record number of foreign national offenders from this country, including a record number of EU criminals.
"Being in the EU gives us access to criminal records sharing and prisoner transfer agreements which help us better identify people with criminal records and, allow us to send foreign criminals back to their home countries to serve their sentences."
The UK is also signed up to the EU's Prisoner Transfer Agreement which means the Government can transfer foreign offenders criminals in prison to their home nations to continue serving their sentence there at their own country's expense.
In the wide-ranging assessment on the work of the immigration services, the committee said there were serious questions over the Home Office's judgment following the "extraordinary" decision it took to deport thousands of people on "questionable or insufficient evidence" of English language testing fraud.
Allegations of widespread fraud in the system lead to heavy-handed treatment of students despite the lack of an independent investigation into the claims, MPs said.
Committee chairman Keith Vaz said: "Despite repeated warnings, the Home Office is still unable to remove foreign offenders from the UK.
"We agree with the Prime Minister that the Home Office should have done better. There are still over 13,000 foreign national offenders in the country, who could fill towns the size of Louth in Lincolnshire, Beccles in Suffolk or Berwick-upon-Tweed in Northumberland, and almost 6,000 of these are living within communities.
"The public would expect our membership of the European Union to make it easier to deport European offenders, but this is clearly not the case, and we continue to keep thousands of these criminals at great and unnecessary expense. These failures are undermining confidence in the UK's immigration system and in the UK's EU membership.
"In stark contrast to these failures are the arrests, dawn raids and aggressive deportations of students from outside the EU. The Home Office appears not to have investigated English language testing fraud allegations themselves before undertaking heavy-handed action. Recent legal cases, with their damming criticisms from senior judges, have opened the door to a mass of expensive and damaging litigation.
"Net migration now stands at 333,000, the second highest figure on record. It will take a modern miracle for the Government to meet its target."
Vote Leave said the foreign offenders were costing £36,000 a year each to jail and that the EU made it "more difficult" to deport overseas criminals.
Backer Dominic Raab, the justice minister, said: "The EU is making us less safe. If we take back control we will be able to deport foreign criminals from our prisons.
"This damning report shows that the EU is making it more difficult to remove dangerous criminals which puts us at risk. It also costs UK taxpayers huge sums of money to keep these people in our prisons rather than sending them home - and this is on top of the £50 million we send to Brussels every day."