Burnham said: "I am in south Wales today because the answers to the UK's challenges will not come from an echo chamber. They will come from the places that know their own strengths, and from a government willing to listen and then act.
"Good jobs in every community, in every part of the UK. That is what I want every leader in this country signed up to."
He will meet the Welsh government's deputy first minister Sioned Williams at the visit.
Williams said the visit would be "a positive example of governments working together to improve people's lives".
She will call for a "fair share of the UK government's steel fund", an "energy social tariff to support those facing fuel poverty and struggling with high energy bills", and high standing charges in Wales to be addressed.
She added: "Wales is an energy-rich nation. People should see that reflected in good jobs, stronger communities and lower bills."
The National Economic Council was created by Gordon Brown's UK government in the wake of the 2008 banking crash, and has now been revived by Burnham.
A previous meeting brought together government ministers and English local authority mayors.
No 10 said the prime minister had wanted to discuss "how we will drive good growth in every postcode across the UK" with the first ministers.
The Welsh government said: "We look forward to discussing shared challenges, including ensuring growth and investment in all the countries represented, and support for families facing the continuing cost-of-living crisis.
"This is alongside further discussions with the UK government to progress broader priorities in Wales, including fair funding and extending devolution."
Plaid Cymru had pressed the UK government for discussions over more powers and funding for the Senedd - Wales' parliament - and the Welsh government in Cardiff.
First Minister Rhun ap Iorwerth had his first meeting with the new prime minister on the sidelines of the Glasgow Commonwealth Games in July.
At the time, ap Iorwerth said it had been "too early" to know whether the new Labour prime minister would agree to his calls.
Downing Street said then that the prime minister wanted an "open and ongoing dialogue" with the first minister.
Welsh Liberal Democrat MP David Chadwick said Wales "needs action" if the prime minister is serious about "easing the burden on Welsh families".
Reform UK's Welsh leader Dan Thomas said ahead of the visit that Burnham "may well be the last Labour prime minister to come to Wales".
"We have no doubt that when the prime minister finally calls a general election and stops running scared, it will be the end of the line for Labour here in Wales," he said.
