The Independent Alliance has sought a meeting with Taoiseach Leo Varadkar on the cost escalations at the Children's Hospital and the ongoing nurses' strike.
That meeting is scheduled for Tuesday amid significant pressure on Health Minister Simon Harris.
Speaking to the Irish Examiner, junior Business minister John Halligan confirmed the Alliance is "backing" Mr Harris and are not seeking his head.
Mr Halligan was scathing in his criticism of attempts by Sinn Fein to call for Mr Harris' removal from office.
"Rather than trying to seek a motion of no confidence in another minister, Sinn Fein would be better served if they went over to Westminster and put down a motion of no confidence in Theresa May, who has the potential to wreck the Irish economy," he said.