SNP ministers were handed new powers over social security following the 2014 independence referendum.
Social Security Scotland now delivers 12 benefits, including a Scottish Child Payment for low-income families.
The move requires complex IT systems to handle new cases and transfer existing data over from the Department for Work and Pensions (DWP).
Digital costs are estimated at £251 million and a dispute has now arisen over funding.
Scottish Labour MSP Pam Duncan-Glancy said: “The last thing we need is more bickering between our two governments.
“The Tories and the SNP keep choosing to pass the buck and escalate conflict instead of working together – and it is holding our social security system back.
“Both governments need to stop playing the blame game and focus on co-operating to deliver a system that works for the people of Scotland.”
Tory MSP Liz Smith said the row was “typical of the SNP’s abject failure to do the most basic tasks of government”.
The DWP’s latest annual report states: “Scottish Government have raised a dispute over their liability to fund certain digital investment activity that DWP believes is essential to deliver devolution.
“The digital investment activity spans a number of years with detailed requirements and costs for future years yet to be defined, therefore a reliable estimate of the contingent liability is not available.”
A spokesperson for the Scottish Government said: “The Scottish Government is working closely with the DWP to resolve this issue under the provisions of the fiscal framework and will not provide commentary while discussions to seek a resolution are ongoing.
“The devolution of social security is a large and complex programme requiring both governments to undertake IT system changes and development.”
Liberal Democrat MSP Willie Rennie said: “This kind of blame game is just unhelpful and tone-deaf to the bigger issues affecting people up and down the country.
“In the midst of a huge cost-of-living crisis, we need governments who are willing to work together, not ones that squabble.
“The Scottish Government needs to step up to the plate and concentrate on delivering the help that people urgently need.”