A meeting with Irish counterpart Micheál Martin was used by Andy Burnham to push for deeper ties between the UK and the European Union.
He emphasised that the relationship with Ireland "matters so much on every single level" as the Taoiseach became the first international leader visiting No 10 North.
The Prime Minister described it as "very fitting" for Mr Martin to be the first head of government to visit the Manchester site due to close ties between the North West and Ireland.
Mr Martin expressed his desire for the UK to build a "stronger and closer" relationship with the EU. Ireland currently holds the rotating presidency of the Council of the European Union.
Mr Burnham said: "The relationship between the UK and Ireland matters so much on every single level.
"Close to home, as co-custodians of the Good Friday Agreement, we are working together to uphold its principles as we approach the 30th anniversary next year.
"In Europe, we are looking forward to a UK-EU summit later this year and, with Ireland in the presidency, wanting to take further steps to deepen the relationship with Europe."
In addition to the visit to No 10 North, Mr Martin was joining the Prime Minister at an Irish club located in his Makerfield constituency.
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Mr Burnham said: "Here in the North West, but across the UK, so many people have Irish connections and they will be so pleased to see you with us today.
"So you are extremely welcome. I’m really pleased that you’re our first visitor here."
Mr Martin noted that it had been a "very challenging and difficult week" for Mr Burnham following the death of his father Roy, adding "I think your sense of duty shines through".
He stated that "in a global world marked by a lot of uncertainty" it remained essential for the two close neighbours to maintain "very strong engagement".
The Taoiseach said: "We look forward to discussing with you even stronger and closer relationships between the European Union and the United Kingdom, which will be of benefit to the European Union and indeed to Ireland, quite frankly."
Mr Martin commended Mr Burnham for his stance on the Israel-Palestine conflict following the Prime Minister resetting the Government’s approach to Israeli Prime Minister Benjamin Netanyahu’s government and stiffening sanctions over West Bank settlements.
He said: "I think for the United Kingdom under your leadership to take the stance that its taken, I think, is important for… standing up for humanitarian values and justice and fulfilling the rights of all in the Middle East to security, to peace and to some decent future for their children and generations to come."
The meeting comes just days after US President Donald Trump created major headlines by stating he would "love to see" a united Ireland while at a meeting with the Taoiseach during a visit in Dublin.
Mr Trump said a united Ireland would "happen eventually" and that the Taoiseach was "the right man over here to get it moving".
