The Irish Government is set to introduce a raft of tighter measures around travel, the Transport Minister has said.
Eamon Ryan said that part of these measures include garda check points at airports.
He added gardaí are stopping people going to and from airports to establish why they are travelling.
The Green Party leader said the decision to set up check points was made because the Government "realised we needed to go further" and tighten up Covid-19 measures.
"That started on Friday - people will have noticed guards' check points at approaches coming out of the airport and (gardaí) are saying 'you can't fly, you cannot be more than five kilometres from your home, you will be fined, you will be prosecuted'. That has to stop immediately," he told RTÉ.
Mr Ryan said there is rising concern among health officials about the new strains of the virus.
"This evening we will be meeting to agree an approach that is being proposed, that we would introduce quarantine measures - that if someone is coming in and they haven't filled in their form in advance of flying, again, sorry you cannot come in - you have to go into quarantine," he added.
Gardai are also set to be given powers to deal with air and ferry passengers who do not have a negative PCR test.
Mr Ryan said that gardaí will be able to prevent them from travelling on into the country.
He also added that people travelling from certain countries, including South American countries and South Africa, can no longer travel to Ireland on a short-term holiday visa.
," he added.
Mr Ryan also said Government will be seeking an all-island approach to travel arrangements in both jurisdictions.
He added: "If we could get an agreement on an all-island basis, if we could get an agreement with the (Northern Ireland) Executive, with the five parties, then absolutely, we will close the door that way.
"If that isn't possible, we will have to work very closely with the UK Government."
The Government will introduce tighter measures around cross-border travel, including garda check points five kilometres from the border.
Mr Ryan added that if anyone is found travelling without a valid explanation they will be subject to prosecution and fine.
"We will significantly increase our surveillance to reduce unnecessary travel," he went on.
Mr Ryan also said that a global response is needed.
"It isn't all just about a ban solves this and the problem goes away, because those people coming in, from say Portugal could easily fly a different route," he added.
Meanwhile, the leader of Sinn Fein said the Government needs to introduce mandatory quarantine for all travellers arriving into Ireland.
Mary Lou McDonald described proposals to quarantine those who arrive without a negative Covid-19 test as "absolutely insufficient".
The Government Covid-19 sub-committee will meet on Monday to discuss the current public health restrictions.
On Tuesday, the Government is to make a decision on whether to extend the health measures in place since last month.
Cabinet will also consider the issue of mandatory quarantine for some arrivals, particularly for those who arrive without a negative test.
However, Ms McDonald said these measures do not go far enough.
"If they don't have a pre-arrival negative test they should be sent back to from whence they came," she told RTÉ Morning Ireland.
"Secondly, we need a second mandatory test after five days of being on the island and we need a mandatory enforced quarantine.
"It seems that the Government is proposing simply to quarantine those that arrive without a negative test - that is absolutely insufficient.
"I'm very surprised at this stage that they don't realise - certainly Nphet and the public health experts realise - we have to grasp this nettle now, and it needs to be very, very clear that only essential travel onto the island, and then a mandatory form of quarantine."
Ms McDonald said that a mandatory quarantine of 14 days will act as an "effective push back" on travel.
The Government is still advising against all non-essential overseas travel.
All passengers arriving into Ireland are required to produce a negative PCR test within 72 hours of arrival.
Passengers are being asked to give evidence of their negative test before boarding a plane or ferry.