Social Democrats co-leader Róisín Shortall has described as "shocking", figures which show that 7% of people are being contacted after arriving in Ireland to check their location.
The figures also show that half of contact calls are being answered.
Speaking on RTÉ's Today with Sarah McInerney, Ms Shortall said the Government is "asleep at the wheel" and has "little or no control" over the status of people travelling into Ireland.
Just over 4,000 follow up calls were made to check up on 60,000 people arriving in to Dublin Airport in the first two weeks in July, according to figures from the Department of Justice and released in response to a parliamentary question from Deputy Shortall.
The figures also show 52% of the 4,121 calls made to passengers were answered.
Deputy Shortall said the Department of Justice is responsible for tracking the movements of people who come into the country, "but they clearly don't want to do that".
She said: "The Department of Justice is waiting for the Department of Health to take over, which they haven't done and this means that there are a significant number of people coming here as tourists and there are little or no checks on their status."
Ms Shortall said the messages from the National Public Health Emergency Team "are stating clearly that they have a significant and growing concern about travel-related introduction of the disease, and the increase in risk of a potential second wave of the virus.
"This is a real danger, and the Government is asleep at the wheel and it is just not good enough when you consider all of the sacrifices people have made here", she said.
Ms Shortall proposed tighter controls at airports and ports through the introduction of a dual testing regime, which would see people tested for Covid-19 on arrival into Ireland, quarantined and tested again on day seven of their stay.
Strict enforcement or supervised quarantine are other measures that could be used to keep better checks on those travelling into the country from abroad, Deputy Shortall added.
"You have to have a system that actually works, not just one that you are paying lip service to."