Taoiseach Simon Harris has called on Ireland's Minister of Justice to table legislation that would allow asylum seekers to be returned to the UK.
Helen McEntee revealed that 80% of recent arrivals in the Republic came from the UK across the Irish border.
Tánaiste (Deputy Prime Minister) Michael Martin said Britain's Rwanda policy was already having an impact on Ireland.
A bill reinstating Britain's Rwanda policy was passed into law on Thursday.
It aims to stop people from crossing the English Channel by sending some asylum seekers to the central African country.
No immigrants have yet been sent from Britain.
The British government had hoped planes would take off by spring, but Chancellor Rishi Sunak said they should take off within 10 to 12 weeks.
According to Irish broadcaster RTÉ, a spokeswoman for Mr Harris said the government had asked Mr McEntee to: I have requested that the revised bill be submitted to the Cabinet next week.” It was reported on Saturday.
Speaking on RTÉ's Six One News, Ms McEntee said: 'There are many reasons why immigration to Ireland is increasing.
“What is clear from the UK's decision to Brexit is that the number of people seeking asylum in their own country is actually increasing, and how to deal with that is British policy.
“My focus as Minister of Justice is to ensure effective immigration structures and systems.
“That is why I am introducing fast-track processing. That is why I will bring emergency legislation to Cabinet this week to ensure that we can effectively bring people back to the UK. To that end, the Home Secretary I am planning to have a meeting with [James Cleverly] I will raise these issues on Monday. ”
A spokesperson for Ireland's Ministry of Justice told BBC News NI that “the issue of irregular movement within the CTA”, the common travel area between the UK and Ireland, would be discussed at the ministers' meeting.
Earlier this week, Mr McEntee told an Oireachtas (Irish Parliament) committee that the number of people crossing the land border between Northern Ireland and the Republic was increasing, and that this now accounted for 80% of the total population. Told. Asylum seeker.
In response to her comments, Mr Martin said the UK government's Rwanda policy meant people were “fearful” of remaining in the UK and were crossing the border into the Republic to avoid being sent to Rwanda. He said he is doing so. Ms Martin, who is also Ireland's foreign secretary, voiced her opposition to the policy.
Speaking on Sky News on Sunday, Chancellor Rishi Sunak was asked whether Mr Martin's comments showed the UK was “exporting problems”.
Mr Sunak said: 'According to your comments, deterrence is already having an effect because people are worried about coming here and this shows exactly what I'm saying. If people come to our country illegally, they know they won't.” If they can't stay here, the chances of them coming are pretty slim. ”
His comments followed reports that around 350 migrants crossed the English Channel in small boats on Saturday. French authorities rescued a further 99 people from three boats lost at sea.
Mr Sunak told Sky News that illegal immigration was a “global problem” and that many countries were seeking to recreate “third country partnerships” similar to the deal struck between the UK and Rwanda. Stated.
Politicians are also discussing cutting net immigration in the UK, with former immigration minister Robert Jenrick calling for a “much more restrictive system” on Sunday.
Writing in the Sunday Telegraph, Mr Jenrick criticized “politicians from all walks of life” for failing to deliver on “promises to control and bring down levels of legal immigration” over the past 30 years, calling for a cap on net immigration. “Tens of thousands of people.
Police Minister Chris Phillip said he was “not in a position to support a hard cap” on immigration levels, but said the government was introducing measures to reduce the number of legal immigrants by “around 300,000 a year”. Stated.
Mr Philippe also appealed to migrants considering making the dangerous Channel crossing from Europe: “Please don't. It's dangerous, illegal and unnecessary.”