A reported rise in migrants aiming to enter Britain using false papers poses real dangers to security, home affairs committee chairman Keith Vaz has said.
EU border agency Frontex reported a 70% year-on-year rise in fraudulent documents held by UK-bound migrants.
This was mostly due to Ukrainians using real Polish ID cards and Albanians with Italian and Greek papers, it said.
The Home Office said passports are required at borders to ensure fraudulent documents cannot be used.
Labour's Keith Vaz called for specialist machines across the EU to check IDs.
"The extent of the fraud revealed in this report poses real dangers for our security and undermines our immigration system," Mr Vaz said.
"When a British immigration officer is presented with an Italian or Greek ID, they are clearly unaware of the history or misuse of that document."
The risk analysis from Frontex says a total of 8,373 "document fraudsters" were detected across the EU in 2015.
It does not specify a separate number for the migrants aiming to enter the UK, but a spokeswoman told the BBC the figure had been 3,945 - up from 2,324 in 2014.
The agency said Syrian, Iranian and Chinese were the other nationalities "often" aiming to reach the UK with fraudulent documents.
Overall there is said to have been a "marked increase" in the number of document fraud incidents recorded on movements within the EU's passport-free Schengen area, which does not apply to the UK.
Mr Vaz said: "Countries which are a source of these fraudulent documents, especially those who aspire to EU membership, must tackle this growing problem, and co-operate closely with Europol and other border agencies across the EU.
"This is simply not happening at present."
Frontex also said terrorists were exploiting the migrant crisis to travel unchecked, and highlighted the security risks associated with the abuse of stolen, blank Syrian passports.
There is no EU-wide system of document inspection performance, it said, adding that sophisticated physical and security features of travel documents present "significant challenges for border-control officers".
Justice minister and supporter of the Vote Leave campaign Dominic Raab said EU law was "forcing us to rely on travel documents issued by other EU countries, and prevents us from issuing our own".
Britain Stronger in Europe said the UK has "full control" over borders because it is not part of the Schengen area, while its deal with the EU allows it to co-operate on security issues.
A Home Office spokesman said: "All passengers entering the UK are checked against police, security and immigration watch lists on arrival at the border.
"Where we believe someone poses a risk, Border Force officers can and do refuse them entry.
"Since 2010, we have denied entry to over 100,000 people, including over 6,500 EU nationals."