Sir Keir has faced criticism for his call for a humanitarian “pause” in the fighting in Gaza as the number of people killed in the enclave continues to rise.
Last month, Mr Burnham joined calls for a ceasefire amid the humanitarian crisis in the region.
Questioned whether his response to the conflict was disloyal to the Labour leader, Mr Burnham told BBC Radio 4's Today programme, the mayor of Greater Manchester said: "No, not at all.
"I would point to the fact that the statement issued by myself, the deputy mayor of Greater Manchester Kate Green, all 10 leaders on a cross-party basis was a careful, considered statement that sought to reflect the views of our communities here in the city region and come to an approach which we think is the right approach.
"Of course we abhor the terrorist attacks of October 7, we recognise Israel's right to take targeted action against Hamas. Our concern was with the widespread bombing causing such a huge number of casualties - that was the point that we have made.
"Keir, in his speech earlier this week, I think made many similar points so I don't think the difference between us is too great.
"And this issue cannot become all about the Labour Party. These are really difficult judgements that everybody is trying to make and I think Keir, the shadow cabinet, the PLP (Parliamentary Labour Party) needs to be given the space to come to those judgements."
Last week, Andy McDonald, MP for Middlesbrough, was suspended after he used the phrase "between the river and the sea" in a speech at a pro-Palestinian rally.
He now sits as an independent MP with a Labour spokesperson saying Mr McDonald's comments at the weekend "were deeply offensive, particularly at a time of rising antisemitism which has left Jewish people fearful for their safety".
A raft of councillors have also resigned from the party after the Labour leader sparked anger by saying Israel had “the right” to cut power and water from Gaza when defending itself from Hamas during an interview with LBC.
In an attempt to quell the rising tensions, Sir Keir later clarified that he did not support Israel committing war crimes, and in a speech earlier this week, he said the country “must submit to the rules of international law” when trying to neutralise terrorists.
“The fact Kier took nine days to clarify what he meant in that LBC interview caused enormous anger,” said one Labour staffer.
“That interview went viral on WhatApp groups. His speech was good, diplomatic, and well delivered but without the call for a ceasefire it was too little, too late for some.”
Both Sadiq Khan and Scottish Labour leader Anas Sarwar have also called for a ceasefire while Shabana Mahmood, the shadow justice secretary, has written to constituents suggesting Israel may be guilty of the “collective punishment” of “innocent civilians” in Gaza. She has not openly spoken out against Sir Keir's stance, however.
Asked whether shadow cabinet members who do call for a ceasefire when Labour’s official policy is for a “humanitarian pause” should be sacked, shadow Foreign Secretary David Lammy said: “Collective responsibility is hugely important and all shadow cabinet colleagues will be reflecting on that at this difficult time.
“This is dividing families. It is upsetting people right across the country and the global community.”
He added: “All of us subscribed to collective responsibility.”