A significant number of infants and children were found buried
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.
"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?" he added.
Earlier, Taoiseach Enda Kenny described the discovery of the mass grave as "truly appalling".
Mr Kenny says the relevant authorities should now be given time to decide how next to proceed.
He was speaking to Midwest Radio in Castlebar.
"Well when this first came to light following the analysis done by Ms (Catherine) 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 subspecies.
"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".
While Minister Leo Varadkar says the commission of investigation needs to continue its work, as so many questions remain unanswered.
Speaking to Newstalk Breakfast earlier, Mr Varadkar said the revelations were not entirely unexpected given the past research into the home by historian Catherine Corless and others. However, he added that "doesn't make it any less gruesome".
He explained: "Ireland in the 20s, 30s, 40s and 50s was a very different place than what it is now - but even by the standards of the time it's abhorrent to me that children weren't given a proper Christian burial.
"You would have expected at the very least a religious order would have ensured they were treated with dignity in death, and that didn't happen."
People Before Profit is calling on the Bon Secours order of nuns - which ran the home in Tuam and others around Ireland - to make a complete and unreserved apology to the victims of the homes.
Dublin TD Bríd Smith called for the order to consider disbanding, suggesting "they have been guilty of a massive cover-up".
Meanwhile, the Coalition of Mother and Baby Home Survivors claims that up to 7,000 babies could have been buried at nine mother and baby homes across the country.
Chairperson Paul Redmond explained that other homes were larger and operating for longer than Tuam.
"Tuam is the tip of the iceberg," he suggested. "It was the fifth biggest of the mother and baby homes...some of them, like St Patrick's on the Navan Road, were four times the size."