The Taoiseach (Irish prime minister) Simon Harris has asked Ireland's justice minister to bring legislation to cabinet to enable asylum seekers to be sent back to the UK.
Helen McEntee has revealed that 80% of recent arrivals to the Republic came from the UK across the Irish border.
Tánaiste (deputy prime minister) Micheál Martin said the UK's Rwanda policy was already impacting Ireland.
Legislation to revive the UK's Rwanda policy became law on Thursday.
It aims to deter people from crossing the English Channel by sending some asylum seekers to the central African country.
No migrants have been sent yet from the UK.
The UK government had hoped for flights to take off by the spring but Prime Minister Rishi Sunak says this should now happen within 10 to 12 weeks.
A spokesman for Mr Harris said the taoiseach had asked Ms McEntee "to bring proposals to Cabinet next week to amend existing law regarding the designation of safe 'third countries' and allowing the return of inadmissible international protection applicants to the UK", Irish broadcaster RTÉ reported on Saturday.
Speaking on RTÉ's Six One News, Ms McEntee said: "There are many reasons why we have seen an increase in migration toward Ireland.
"What's clear in the decision that the UK have taken in choosing Brexit, they have actually seen an increase in people seeking asylum in their country. The way that they deal with that, it's their policy.
"My focus as minister for justice is making sure that we have an effective immigration structure and system.
"That's why I'm introducing fast processing. That's why I'll have emergency legislation at cabinet this week to make sure that we can effectively return people to the UK and that's why I'll be meeting with the Home Secretary [James Cleverly] to raise these issues on Monday."
A spokesman for the Irish Department of Justice told BBC News NI that "the issue of irregular movement within the CTA" - the Common Travel Area between Britain and Ireland - would be discussed at the ministers' meeting.
'Fearful'
Earlier in the week, Ms McEntee told a committee of the Oireachtas (Irish Parliament) there had been a rise in the number of people crossing the land border between Northern Ireland and the Republic, with this now making up 80% of the total number of asylum seekers.
In response to her comments, Mr Martin said the UK government's Rwanda policy meant people were "fearful" of staying in the UK and were crossing the border to the Republic so they would not be sent to Rwanda.
Mr Martin, who also serves as Ireland's foreign minister, has spoken of his opposition to the policy.
A No 10 spokesperson said it was "too early to jump to specific conclusions about the impact of the Act and treaty in terms of migrant behaviour" following Mr Martin's comments.
The Safety of Rwanda Act, which aims to avoid further legal challenges to the policy by declaring Rwanda a safe country, was approved by MPs and peers this week and passed into law on Thursday.
However, the plan could still be held up by court challenges.