Spread equally across all member states, this would bring some 2.8m (2.4m) to Scottish milk producers. The union said the best, easiest and simplest method of distribution of this money would be to pay it direct to Scotland's dairy farmers.
It has been reckoned that the Scottish share of the EU money will work out at about 0.2p per litre, and if the proposal is adopted by the Scottish Government, the payment to the average Scottish dairy farmer would be about 2,000.
After a meeting on the issue, milk committee chairman Jimmy Mitchell said: "While the sums of money involved are relatively small, every little bit helps, as the majority of dairy farmers are still struggling to achieve an acceptable and fair price for their milk."
The additional cash was agreed in Brussels in November following demonstrations across Europe over the low prices producers were receiving for their milk. It was left to member states to decide their own priorities on how the cash was allocated.
While welcoming the additional money, Mitchell described it as a "bit of a sticking plaster". It would not deflect the union's attention away from its wider agenda of securing a better milk price from the marketplace for our members, he said.
"Wholesale prices for cheese, butter, cream and milk powder remain significantly higher than in the same period a year ago, and demand for fresh liquid milk remains buoyant.
"Despite these positive signs, very few milk producers in Scotland have seen any significant change to their milk price in recent months, and we continue to meet and discuss with Scotland's milk purchasers, highlighting the urgent need for farmgate prices to improve."