It comes a year after the United Nations Convention on the Rights of the Child (UNCRC) Bill was passed unanimously by MSPs in Holyrood.
The Bill, which sought to incorporate the convention into Scots law, was subject to a referral by the UK Government to the Supreme Court due to concerns it went beyond the powers of the Scottish Parliament.
The Scottish Government lost the court case after judges ruled the Bill would undermine the Scotland Act.
Ministers have since faced pressure to bring the Bill back to Holyrood with the appropriate changes. However, the Scottish Government is seeking to change the devolution settlement before doing so.
Letters between John Swinney and Scotland Secretary Alister Jack state “significant changes” to the settlement are being sought by the Scottish Government.
Children’s minister Clare Haughey has also said the Government is examining “extending our powers” to incorporate the UNCRC beyond the existing devolution agreement.
Willie Rennie, the Scottish Liberal Democrat MSP, said another “row” with the UK should not be the priority for the SNP.
He said: “The Supreme Court was clear about which changes can be made by the Scottish Parliament. Yet a year on the SNP/Green Government has still not come up with a fix, instead exploiting this issue to re-run old arguments over where powers lie and have a political fight with the government in Westminster.
“The SNP’s track record on children's rights legislation is rather meagre and certainly insufficient. They seem to care more about constitutional squabbling than they do the rights of young people.
"A year ago Parliament overwhelmingly backed this legislation and we need to see it return to Parliament urgently with fixes in place.”
The letters between the two governments state the Scottish Government has “identified potential routes to increasing the effectiveness of incorporation, beyond those that are now available to the Scottish Parliament alone”.
The letter from Mr Swinney to Mr Jack continues, adding this would include ensuring UK-wide acts in devolved areas such as education were “meaningfully subject to international human rights standards”.
In his response, Mr Jack states: “I feel it is important to be clear at the outset that the UK Government’s position is that the current devolution settlement strikes the right balance, and that amendments to the Bill and any other proposals would need to be within the parameters of the devolution settlement as it stands.”
A Scottish Government spokesperson said: “We remain committed to the incorporation into Scots law of the UNCRC to the maximum extent possible as soon as practicable.
“It is vital that we work through the complex issues raised by the Supreme Court judgment to ensure that incorporation can happen as quickly as possible with confidence that any amendments to the Bill do not attract further challenge. The majority of work in relation to implementation of the UNCRC is continuing.
“One aspect which we have been pursuing is engagement with the UK Government to explore whether powers under the devolution settlement can be used to give the UNCRC Bill greater effect than is possible under the Scottish Parliament’s current powers.”