Calls for the devolution of policing and criminal justice had been shared by Plaid Cymru and the Senedd’s Welsh Labour politicians, but not the previous UK Labour government led by Sir Keir Starmer.
In January, Home Secretary Shabana Mahmood said plans to overhaul policing services would not see the Welsh government given new powers.
Burnham, who succeeded Sir Keir in July, made the comments at the Port Talbot steelworks, on his first official visit to Wales since he entered No 10.
Labour had been unseated from the Welsh government for the first time in its history by Plaid Cymru in the Senedd election in May.
Burnham met Welsh government Deputy First Minister Sioned Williams in Port Talbot on Thursday, with First Minister Rhun ap Iorwerth currently on leave.
Ap Iorwerth made negotiations with Westminster for more powers a key part of his election campaign.
Burnham said he had promised the Welsh government “to look at any asks for further devolution and policing has been specifically mentioned within that”.
“That’s an issue I’m certainly prepared to look at, and I hear what you’re saying,” he said.
“As a former mayor that had those responsibilities, I understand how much you can do when you can really make policing right for the places and the people where you are.
“So I hear that call, I’m open to the discussion with the Welsh government
“As part of the wider policing reform that’s going on, it seems a good time to have a look at those issues.
“So I can’t promise today, but what I can say is I’m open to the conversation.”
The Press Association reported that Wales’ appeals to get control over the Crown Estate – revenue raised from land owned by the monarch – would be given a “fair hearing” by Burnham at the October summit.
“Maybe there’s an argument for deeper devolution, so more power into the cities and the regions of Wales, to north Wales working more with the North West, or mid Wales more with the West Midlands, of the kind of devolution that is developing now quite quickly across England,” Burnham said.
In a book two years ago Burnham said it had been time to “tear up” the Barnett Formula funding arrangements.
Wales reported in June that Burnham had no intention of reforming the system that allocated cash to Wales, Scotland, and Northern Ireland based largely on their share of population.
Critics have argued that it did not fully take Wales’ needs into account.
In Port Talbot, Burnham indicated that reform of Barnett was still not on the table.
He said: “I have a very clear view about creating a Britain that’s fair, that treats all people and all places equally.
“I’m well aware of the concerns about the Barnett formula here in Wales.
“I have to be honest and say – there’s only so many things I can do in the time that I’ve got.
“At the moment I can’t promise there will be moves to review that.”
Burnham’s comments on policing prompted concern from one Labour MP, who told Wales: “It will be a travesty and not in Wales’s interest.”
They said it would not be “welcomed by a majority” of Welsh Labour parliamentarians in Westminster.
Another Labour MP said it was “good” that the prime minister was open to the devolution of policing, and there was “no harm” in discussing it.
But they added: “My feeling is they shouldn’t devolve anything until we’ve got a grip of the NHS and education and Plaid have earned their stripes.”
Plaid’s Saville Roberts said devolving policing to Wales would “help keep our communities safer”.
She added: “It is also welcome to hear that the prime minister is open to considering the case for devolving the Crown Estate, which Plaid Cymru has long argued would be a crucial step towards unlocking greater economic growth for communities across Wales.”
Sioned Williams, deputy first minister, said before her meeting: “We’re hearing the right noises around the importance of devolution and taking that power away from London and out into the regions and nations of the UK.”
She said the announcement of the National Economic Council summit was an “hopeful thing” but could not be a “talking shop”.
Reform Wales Leader Dan Thomas objected to the devolution of policing, adding: “Plaid’s record on devolution, in cahoots with Labour over the past 27 years is one of failure and chaos.
“Wales has a failing health service, falling education standards and a struggling economy. Can you imagine what Plaid will do to policing here in Wales?”