But the dynamics in Wales are different from those in Scotland.
Ap Iorwerth, who was appointed the first non-Labour first minister of Wales after May’s Senedd election, has ruled out pushing for a referendum on Welsh independence during his party’s first term in government, saying he wanted to lead a “national conversation” about the issue.
His government intends to set up a national commission to “lay the foundations” for future plans for Welsh independence.
In the meantime, Welsh ministers are seeking extra powers to be transferred to Wales, including over policing and youth justice.
Burnham has said he is “open” to the idea.
In Northern Ireland, the first and deputy first minister have identical powers and responsibilities and the Democratic Unionist Party’s (DUP) Emma Little-Pengelly is the deputy first minister.
Little-Pengelly criticised O’Neill for attending Monday’s talks.
The talks are likely to cover the way the devolved nations are funded by the UK government.
Welsh ministers have long called for the Barnett funding formula to be reformed, claiming it disadvantages Wales.
However, it is argued Scotland does well from the formula.
Before he became prime minister, Burnham made the case for change, but on his recent visit to Wales he played down the prospect of reform.
The devolved leaders will meet the prime minister next month.
The UK government said: “The prime minister is committed to delivering change across all four nations of the United Kingdom and believes strongly in the union.”
Welsh Conservatives leader Darren Millar said: “Instead of obsessing about independence, the first minister should be focused on dealing with the crisis engulfing the Welsh NHS and the declining standards in Welsh schools.”
Reform UK Wales said: “Plaid Cymru is showing its true colours by meeting with Sinn Féin and the Scottish separatists.”