Ms Burton is ruling out any proposed alternatives to cutting the respite care grant, as some Fine Gael TDs scramble to find different savings.
Several Fine Gael backbenchers are understood to be drafting alternative proposals to the cut for Ms Burton to consider, but these will not be entertained by the minister.
The Fine Gael group is understood to include Wicklow's Simon Harris and Meath East's Regina Doherty. But Ms Burton has refused to meet with them.
A number of Fine Gael TDs are concerned about the cut, which will see the annual grant fall by €325 from €1,700 to €1,375.
Sources close to the minister have ruled out adopting alternatives that would save the same amount of money, €26m a year.
A Coalition source said: "It's a package and it is not being unravelled. We are not looking at reviews or alternative. The Budget is a package, as has been said on numerous occasions by the Taoiseach, the Tanaiste and ministers."
Fine Gael TDs are annoyed by Ms Burton's behaviour, and the matter is expected to be thrashed out at a party meeting tonight.
A party TD said: "Labour is supposed to be the conscience of the Government, but the minister is refusing to engage. There is widespread dissatisfaction and anger with Joan Burton and it is likely to come up at the Fine Gael parliamentary party meeting.
Another deputy said: "This wasn't a very reforming cut. It was a blunt one. Where is the caring Labour Party?"
"It is very disheartening when the people you represent have their views ignored," said a backbencher. "I am sick to death of being told we are part of the big, bad, right-wing Fine Gael party."
The Government is ruling out any rowback on the social welfare cuts outlined in the Budget last week.
Last night, Ms Burton began the passage of the legislation to bring the cuts into effect.
Highlighting the protection of the core social welfare payments, she said: "For the second year in a row there has been no cut in weekly payments, which is a considerable achievement."
Fianna Fail leader Micheal Martin vowed to force Government backbenchers to vote specifically on each reduction in social welfare.
He said the cuts to respite care grants were "a slap in the face to these individuals", adding that it sends out a message that is "appallingly mean-spirited and contemptuous".
Defending the Labour Party from Mr Martin's jibes, Taoiseach Enda Kenny said the attack on his coalition partners was "utterly hypocritical and opportunistic".
"You're under pressure from your colleagues on the far side of the House," he said, nodding towards Sinn Fein.