Deputies are demanding that decisions such as cuts to child benefit and the respite care grant be "re-examined" by Ms Burton and her officials this weekend.
There is growing anger over the respite care grant which will save the Government an estimated €24m.
There had been some hope that the Government would u-turn on some measures after it rowed back on education cuts following last year's Budget.
Meetings between TDs and officials in the department continued today, with sources close to the process admitting that there was little hope of any changes.
"We're being told the decisions have been made and that's the end of it," a well-placed source said.
Labour chairman Colm Keaveney is understood to be adamant that scope exists to make changes to the measures before they are ironed out in the Dail.
The Galway TD, who has proven a constant thorn in the side of the Labour leadership, tweeted that he can only live with "elements" of Wednesday's Budget.
A date for the passing of the Social Welfare Bill has not yet been fully decided, leaving the door open for last-minute amendments.
However, a number of sources in the Labour party are dismissing Mr Keaveney's stance as a "publicity stunt".
Other party TDs understood to have serious misgivings about elements of the Budget include Clare TD Michael McNamara and Dublin TDs Eamon Maloney and Michael Conaghan.
Mr Conaghan told the Herald this morning that the cut to child benefit is a "huge issue" for him and that he met department officials to outline his concerns. "People elected me to try. There are parts of the Budget that I am unhappy with and I will do all I can to express this point of view. You never know what you achieve by doing this. Sometimes you have good outcomes and sometimes you don't but if I didn't try I'd be failing those who elected me."
Asked whether he believed there would be changes to the Budget, Mr Conaghan said: "You never know but people expect you to try."