Brussels has demanded Britain give a key role in its fight to stop Channel small boats to EU judges as the price of more help with the migrant crisis.
Andy Burnham will use a summit next month designed to draw the UK closer to the bloc to get a new deal that includes access to its migrant and criminal databases that was lost after Brexit.
However, the EU is said to be insisting the UK give oversight of its use of data to the Luxembourg-based European Court of Justice (ECJ), as well as paying a fee.
The ECJ already has a controversial role in overseeing the Windsor Framework that governs the border between Northern Ireland in Ireland under a deal struck by ex-PM Rishi Sunak.
But any further derogation of powers is likely to spark fury among Brexiteers, who already believe that EU countries are failing to pull their weight over migrants.
Tory deputy leader Alex Burghart told the Telegraph: 'If access to EU databases means handing over taxpayers' money and accepting ECJ oversight, Labour must explain the price.
'You cannot put a price on sovereignty.'
Andy Burnham will use a summit next month designed to draw the UK closer to the bloc to get a new deal that includes access to its migrant and criminal databases that was lost after Brexit
However the EU is said to be insisting the UK give oversight of the scheme to the Luxembourg-based European Court of Justice, as well as paying a fee
The EU last year rejected Sir Keir Starmer's bid to access the bloc's crime and illegal migration databases.
The Schengen Information System gives EU member states real-time alerts on criminal suspects.
There is also Eurodac, a fingerprinting system with a record of all illegal migrants caught trying to cross into the EU without permission.
Mr Burnham was asked about his Brexit reset in the Commons on Monday, telling Labour MP Chris Webb: 'One of the most positive aspects of my meeting with President von der Leyen in New York was her openness to the conversation about restoring data sharing on immigration across Europe.
'That is something that Britain has long sought in the decade since Brexit. It is the control that we lost, actually, at the time of the referendum.
'It is one issue that does not get enough attention and it should get more. I say to my hon. Friend that I will work hard to see if we can get a practical arrangement on that very issue at the summit in November.'