The breakthrough would make it easier to extradite criminals and catch terrorists after the transition period has ended. So far Britain rejected demands from the EU to remain part of the European Convention on Human Rights (ECHR).
The move halted any progress in reaching agreements on intelligence sharing and a possible new treaty to replace the European Arrest Warrant system.
Brussels claimed that the commitment was legally crucial for EU countries to surrender criminals or share criminal data.
The UK government told the EU that they would not welcome a deal that controlled the implementation of the ECHR in British law.
The UK also insisted they had no plans to abandon human rights law.
The ECHR is an international convention to protect human rights and political freedoms in Europe.
It is drafted by the Council of Europe and enforced by the European Court of Human Rights in Strasbourg, France.
The Council of Europe is an international organisation who aims to uphold human rights, democracy and rule of law in Europe but is not an EU institution.
It includes countries such as Russia and Turkey, and the UK did not leave the Council of Europe when Brexit happened on 31 January.
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On Monday, Prime Minister Boris Johnson met virtually with the three presidents of the major EU institutions.
The meeting with the President of the European Commission Ursula von der Leyen, President of the European Parliament David Sassoli, and President of the European Council Charles Michel gave some optimism.
Mr Johnson echoed his commitment to the ECHR in the meeting.
On Friday, Ms von der Leyen said: “No one can say with certainty where these negotiations will be at the end of this year, but I know for sure that we will have done everything to reach an agreement.”
A UK source close to the negotiations told the Telegraph: “We simply had a more constructive discussion of the reality of our commitment to human rights law.
“We come from different positions on this but behind it is a bit of a question of good faith.
“We are a good faith member of the ECHR and we tend to be so. And we think that should be good enough.”
The source continued: “Clearly there’s still a big, big difference on this question of our domestic legislation. And our right to evolve it.”
ECHR rules are incorporated into UK national law in the Human Right Act 1998.
When serving as Home Secretary, Theresa may said that the UK should quit the ECHR as it made it harder to deport some criminals.