London said it wanted to maintain existing partnerships with a bespoke pact that would open the door to enhanced cooperation as terror threats evolve against British and EU citizens.
The government said it was "vital" for public safety throughout Europe that Britain and the EU maintain and even enhance close collaboration after the UK leaves the bloc in March 2019.
A paper on the matter examined existing partnership agreements with countries outside the European Union but said they were not suitable because they would fall short of current capabilities.
"Developing a new framework to sustain this cooperation will require a shared level of ambition: the UK and the EU need to look beyond existing third country precedents," the paper said.
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It said Britain should be able to reach an agreement with Europol, the EU's criminal intelligence agency, providing the same level of cooperation as now.
London is not ruling out staying inside Europol and the European Arrest Warrant system.
"The UK brings leading capabilities and expertise in security, the delivery of justice and the fight against crime and terrorism," the paper said.