Taoiseach Micheál Martin has told the Dáil that "we have to be vigilant" over abuses of the asylum process.
"I think there are issues there that you have highlighted", Mr Martin told a deputy, with "people coming from some EU countries with documents that were dispensed of when they arrived here, and then applied for asylum".
He was responding to Independent TD Matt Shanahan, who said that official figures reveal that this year 3,705 people had arrived in Dublin Airport without documentation up to the end of September.
Of a total of 5,662 people who were refused entry to the State, 4,969 intended to claim asylum, he noted.
"It is clear that a significant number hold status in another member state," he added, and urged that procedures at borders be tightened.
Meanwhile, the National Co-ordinator of the Ukraine Civil Society Forum said there is a growing concern over objections to refugees arriving in Ireland.
Latest Government figures suggest that by the end of the year there could be 72,000 Ukrainians seeking temporary protection here. About 55,000 will need accommodation.
At the same time, there has been a rise in the number of people from other countries seeking asylum in Ireland.
Speaking on RTÉ's Morning Ireland, Emma Lane-Spollen said there is "growing concern" over objections to refugees arriving.
"Our response so far has been extraordinary and people have shown enormous goodwill. They've opened their houses, and they've extended huge friendship and people have felt huge gratitude. Ukrainians have been so surprised about how warm Irish people have been to them and that friendliness.
"But there is growing concern and we're hearing it through our network of family resource centres and others. Just an uneasiness happening."
She said there is a need for "much better communication and a real need for better planning so communities understand what's happening and that the Government has a plan."
She said that "fundamentally communities don't like to feel in the dark. They don't like to feel that thing is being done to them and they may be hurting for many reasons. Often there are significant structural problems and resource issues that need to be listened to. But those crises are not caused by refugees and they won't be solved by shouting or intimidation like we saw at the East Wall over the weekend".
Ms Lane-Spollen said it is not clear whether the numbers of people coming from Ukraine has been affected by the accommodation issues here.
She said the deteriorating situation in Ukraine with Vladimir Putin "weaponisng energy, he's weaponising food, he's weaponising migration and his goal is to weaken and destabilise European countries across the board and we must not play into his hands."
The next six months will be really hard, she said, particularly hard Ukraine, but it is also going to be tough here.
"There is no sugarcoating that we are going to have to work together, but it is about working together. And falling back on the national meitheal that is there in Ireland. That is who we are and I think we really need to kind of channel that to get us through these very tough times. "
Ms Lane-Spollen said that responsibility for refugees should be reallocated to the Office of An Taoiseach.
"It's the only way because you need a whole of government response and the Taoiseach is the only one who can really demand that.
"And it's not just about co-ordinating, it's about driving because the solutions that have been put on the table so far are not ambitious enough."
The former Secretary General of the European Commission and member of the external advisory group on ending direct provision said Ireland needs to "gear up" and plan for a long-term future of including refugees into society.
Speaking on RTÉ's News at One, Catherine Day said people will always have cause to flee and will want to come to safe havens like the EU and Ireland.
She said the State needs to provide reception centres built on State-owned land and that it should adopt emergency powers so it does not have to go through the planning system.
"We need to engage local communities, where there is enormous goodwill, but where people want reassurance that if there are suddenly more children in school or more people needing the local GP, that extra services will be provided because we have a bigger population," she said.
Ms Day said it is "very important" that local communities do not feel decisions are taken over their heads and that they understand why people need to be accommodated and what the impact will be.
Communities should not have veto on issue of housing refugees
The Tánaiste has said it is important that communities are consulted about the housing of refugees and asylum seekers, but they cannot have a veto on the issue.
Leo Varadkar said the Government needs to work out how it can carry out consultation better in the future.
"But I don't think any community can have a veto on who gets to live in their area," he said to reporters in Dún Laoghaire this morning.
"It has never been the case that when a new housing estate was built near me that I was consulted on who got to live there. The same thing would apply to a new apartment block.
"And I think we need to be very careful not to make the mistake of confusing consultation and information with communities, which is important, with the idea that any community can have a veto on the kind of people who get to live in their area. That’s not right."
Mr Varadkar said he understand why people call for certain actions or projects, like the challenge of accommodating Ukrainian refugees, to be taken in under the Department of An Taoiseach.
But he said it is not the case that the Taoiseach’s department is the largest. He said it has 150-200 staff and a budget which is less than a tenth of the Department of Children.
He was speaking following calls for the issue of accommodation of refugees from Ukraine to be brought under the control of the Department of the Taoiseach in order to achieve more coordination.
"So I think sometimes people believe that by putting something under the Department of the Taoiseach that it will make things happen," he said.
"That is a misunderstanding of the scale and the resourcing that the Department of the Taoiseach has."
He added that where the Taoiseach’s department does come into play is in coordination and ensuring that every department "does their bit" when it comes to dealing with the problem.
He said that is the role that department has and is taking.
The Tánaiste said people who come here from Ukraine are welcome and the Government will do everything it can to provide them with shelter, including in people's homes, retrofitting office blocs, rapid build and modular housing.
He said it is impossible to make an absolute commitment on the type of accommodation that Ukrainian refugees will receive.
"I would love to be able to say that we can provide own door self catering accommodation to everyone who arrives here from any part of the world, I can’t promise that, and I don’t think any Government in the world is able to do that," he added.
Additional reporting Will Goodbody