"It's not acceptable to me that here there is a live investigation taking place and we cannot have information being put in the public domain that's not in the direct control of British police and security service," Burnham told the BBC Wednesday.
"To have information put in the public domain before it was put there by people here is just wrong."
Twenty-two people were killed in Monday's attack at the Manchester Arena, when a 22-year-old man detonated an explosive device shortly after an Ariana Grande concert.
British Prime Minister Theresa May is likely to meet with US President Donald Trump during Thursday's NATO summit amid the stern reaction from UK authorities to the regular leaks.
When asked about the leaks, NATO Secretary-General Jens Stoltenberg said that sharing intelligence was "of great importance. Sharing intelligence is based on trust."
Earlier Britain's National Police Chiefs' Council warned that leaks of potential evidence "undermines our investigations." Home Secretary Amber Rudd said they were "irritating," adding she had made it clear that it "should not happen again."
Within hours of Rudd's statement, The New York Times
published photographs purportedly showing remnants of the bomb detonated at the concert. They purported to show a detonator, battery, nuts and screws for shrapnel, and fragments of a backpack used in the attack.
A Greater Manchester Police spokeswoman issued a very firm "no comment" to CNN on the photo publication. The Times, without specifying the source, said British authorities provided access to photos of materials found at the scene.
Authorities value the trusted relationships they have with partners around the world, a spokesperson for the National Counter Terrorism Security Office said.
"When that trust is breached it undermines these relationships, and undermines our investigations and the confidence of victims, witnesses and their families," the spokesperson said. "This damage is even greater when it involves unauthorized disclosure of potential evidence in the middle of a major counter terrorism investigation."
'(Leaks) should not happen again'
A string of details about the deadly attack
have emerged from US law enforcement sources before being released by British police or officials.
"The British police have been very clear that they want to control the flow of information in order to protect operational integrity, the element of surprise," Rudd told BBC Radio 4's "Today" program. "So it is irritating if it gets released from other sources, and I have been very clear with our friends that that should not happen again."
Asked whether the leaks from US officials had compromised the ongoing investigation, Rudd said she "wouldn't go that far."
But, she added, "I can say that they are perfectly clear about the situation and that it shouldn't happen again."
The UK terror threat level has been raised to critical
for the first time in over a decade in the wake of the bombing -- meaning more attacks may be imminent -- as investigators seek to track down any associates of the suspect, named as 22-year-old Salman Abedi
Rudd earlier told Sky News that she expects the critical threat level to be temporary. She also said the bomber was known "up to a point"
by the intelligence services.
Abedi is believed to have died in the powerful blast, though he has not yet been formally identified by the coroner, Manchester police said.
The United States is one of Britain's key intelligence partners. Both countries share sensitive information as part of the "Five Eyes" alliance, which also includes Canada, Australia and New Zealand.
Details were quick to emerge from the US in the immediate aftermath of the bombing. For example, US officials told CNN that the attack appeared to be a suicide bombing and that a male at the scene had been identified as the probable suicide bomber.
The US government has come under scrutiny over its handling of foreign intelligence
in recent days following reports that US President Donald Trump shared top secret information originating from Israel during a White House meeting with senior Russian officials.
Price of dealing with US
Shashank Joshi, a senior research fellow at the London-based Royal United Services Institute think tank, told CNN the leaks of information from the US side seemed to be more frustrating than catastrophic to the Manchester investigation.
"A lot of the information that leaked overnight Monday was fairly mundane, about casualty figures and the method of attack, but the leaking of the suspect's name was more disruptive because it might have tipped off other suspects," he said.
Keeping the suspect's name quiet for longer might have given the police more time to chase up other leads that would otherwise have shut down, he said.
"It's partly a function of modern counter-terrorism that places a premium on rapid sharing of information across borders in fast-moving investigations," Joshi said. "That has its advantages, such as being able to piece together bits of information from different countries in a way you wouldn't have been able to do 25 years ago. And it has disadvantages, like exposing your own information to more places where it can leak."
Joshi noted that the Manchester information seemed to have come from US law enforcement, rather than security agencies.
It's not the first time the US has faced criticism for leaking information on a UK terror investigation, Joshi said, citing the naming of the Kuwaiti-born Londoner Mohammed Emwazi
, a ISIS fighter also known as "Jihadi John."
Part of the issue is the sheer size of the US national security bureaucracy, which means there are more points from which information can leak, he said.
"That is the price, in a way, of dealing with world's most capable national security operation," he said, adding that the US is "by far and away" the most significant intelligence partner for Britain.
Stern words will likely be directed to the US side, he said, but "on balance, it's probably not going to change intelligence-sharing arrangements all that much."