Hundreds of thousands of migrants entered the country through the points-based system.
Home Secretary Shabana Mahmood has sought to crack down on migration by extending the minimum period of time required to apply for indefinite leave to remain (ILR) from five years to 15.
But she risks sparking a Labour civil war and may yet water down her migration reforms.
Energy Secretary Miatta Fahnbulleh declined seven times in a Sky interview to say whether she agreed with extending the period for ILR, later saying it was “awkward” because she had promised to “never lie” when entering politics.
She later backed down on social media, saying: “I fully support our Home Secretary and the Government’s position on migration reform”.
Mr Burnham is also currently considering different “options” to bring the UK closer to the EU, including joining the single market, customs union, maintaining the current status quo, or rejoining the bloc entirely.
The long-delayed EU-UK summit is set to take place on November 20 in Brussels, with discussions to be held on “the defence and security partnership, energy, agrifood rules, climate, people-to-people contacts, and migration and law enforcement”, according to the bloc.