They claim that under Barnett formula rules, Wales should get an extra £1.67bn and Scotland £2.9bn.
A letter has been sent to the UK Treasury to trigger the dispute resolution process.
UK ministers said the deal was made outside of the formula but was part of supporting growth across the UK.
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 and his Scottish counterpart Derek Mackay have insisted Barnett rules should apply in this case too.
"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," he said.
Responding, a UK Government spokesman said: "As we have made clear, this agreement is part of the government's commitment to support growth across all parts of the UK.
"Like previous Northern Ireland support packages, and the funding of over £1bn for City Deals in Scotland and Wales, this funding is a targeted intervention to address a specific set of unique challenges.
"And as was the case for those previous interventions, this exceptional funding will be made outside of the Barnett funding system."
The Welsh and Scottish governments have raised a formal objection to Theresa May's £1bn deal with the Democratic Unionists to keep the Tories in power.