A member of the Cabinet says the Commission of Investigation into Mother and Baby Homes must be allowed finish its work in relation to the former Bon Secours facility in Tuam.
Last week, the commission confirmed it had discovered the remains of a "significant number" of children in a sewer system at the Galway site.
Politicians on all sides of the political divide have condemned the find, which comprises of hundreds of discarded babies' bodies.
The home in Tuam operated between 1925 and 1961, and the samples are likely to date from the 1950s.
It is believed several hundred children were buried at the site.
Advocacy groups have been calling for similar excavations at other former homes around the country.
Public Expenditure Minister Paschal Donohoe says it will be considered.
He said: "Cabinet and Minister (Katherine) Zappone will of course consider whether there is the potential for any such discoveries elsewhere within our country.
"But before we go ahead with any such decisions, I think it is important to establish do we have any evidence that might prompt such excavations?"
The Taoiseach Enda Kenny has described the discovery of a mass grave at the Tuam Mother and Baby Home as "truly appalling".
Politicians on all sides of the political divide have condemned the Galway find which comprises of hundreds of discarded babies' bodies.
Mr Kenny says the relevant authorities should now be given time to decide how next to proceed.
Speaking to Midwest Radio, the Taoiseach said: "Well when this first cam to light following the analysis done by Miss Corless, the local historian in Tuam, I described the way that babies of single mothers were treated in this country, back in 2014, as being akin to some kind of sub-species.
"It's appalling, truly appalling."
"And obviously the coroner and everybody involved has to see how best we can proceed with the next step in this case, and possibly others.
"It's a horrendous situation for those whose siblings were treated in this fashion."
Asked if the Commission of Investigation into mother and baby homes needs to be extended following the Tuam discovery, he said: "When the Minister for Children's talking to the commission, obviously if it needs to be extended then it will be extended."
"But I think the next step, if I understand this, is for the coroner to move here - but the question is what do you do to attempt to identify the remains of a substantial number of babies between three weeks and three years (old) which have come to light in this case.
"And are there others, in other locations, who were treated in the same fashion?
"So this is another issue - one of many that we have come across in the last number of years - which were left lying in the shadows of an Ireland that we had hoped was gone"
"But I commend the local historian, Ms Corless, who followed through on this - and now it's beginning to come to light the scale of what actually happened".
The Bon Secours order that ran the Tuam mother and baby home has come under pressure to disband and to give up their assets to the State, .
People Before Profit (PBP) have called on the nuns to make a complete and unreserved apology to the victims of the mother and baby home.
Speaking this morning AAA-PBP TD Bríd Smith claimed the Bon Secour nuns had been guilty of a "massive cover-up" of "criminal activity" which they had denied for many years.
Ms Smith was joined by Deirdre Wadding, a former resident of Bessboro mother and baby home in Cork.
Ms Wadding said the revelations that more than 700 babies had been buried in pits in the Co Galway home, had shocked her but proved that "church and State colluded from the very foundation of this State to oppress women. Women and children have been brutalised by those twin forces for decades now.
"For me it has been a very wobbly few days, to be personal about it, as I read about Tuam it certainly stirred up my own experiences," said Ms Wadding who was sent to Bessboro aged 19 in 1981.
She said even then the culture of "shame, isolation, banishment and loss" still existed.
Calling on the order to give up their assets and disband Ms Smith said: "Bon Secour now run, and make lucrative profits from two private hospitals, one in Glasnevin and the other in the University area of Cork. They are a very wealthy organisation."
She added that their resources should be used to compensate families involved and to provide memorial services to remember the children buried in Tuam and other homes.
"The order itself should be disbanded and their assets handed over to the State.
"They would be two fine hospitals that the State could take over and run, not for profit but for the benefit of everybody."
"We need to once and for all tell the church to get out of our lives, get out of the lives of our children, get out of the lives of our women, get out of beds, get out of our schools and get out of our hospitals" Ms Smith said.