Under Barnett formula rules, Wales should get an extra £1,670 million, and Scotland £2,900 million.
A letter has been sent to the UK Treasury to trigger the dispute resolution process.
The Barnett formula is used by the UK Treasury to determine how much extra money should go to Wales, Scotland and Northern Ireland in the event of an increase in public spending in England.
Wales' Finance Secretary Mark Drakeford MA and his Scottish counterpart Derek Mackay MSP insist Barnett rules should apply following the Tory deal with the Democratic Unionist Party (DUP) to stay in power at Westminster.
"The UK Government has abandoned these well-established arrangements to the detriment of Wales and other parts of the UK," Mr Drakeford said.
"It is simply inexcusable that the UK Government is willing to 'bypass' those rules."
He added: "At a time when public services in Wales are under pressure as a result of the UK Government's damaging and ongoing policy of austerity, it is only right that Wales gets its fair share of funding through the established rules of the Barnett formula."
Mr Mackay said the Scottish Government "fundamentally disagrees with the way in which this additional funding for Northern Ireland has been allocated".
"It remains my hope that we will be able to reach a satisfactory resolution to this situation which results in a funding allocation across Scotland, Wales and England that is fair and reasonable for all - but in order to reach such a solution we now need to pursue the formal dispute resolution process."
Added Kenneth Gibson:
"The Tories' desperate attempt at holding on to power by buying off Northern Ireland's DUP upsets public sector funding arrangements across the UK.
"The people of Scotland, Wales and indeed England have been short-changed by this austerity busting deal. Each nation should receive the funding share previously agreed under the Barnett formula."