Health minister Robin Swann has asked his cross-border counterpart to consider new laws and data-sharing agreements to help track international travellers arriving on the island.
Mr Swann has written to Stephen Donnelly expressing concern about the "inability" of both jurisdictions to capture and share information about passengers transiting through one part of the island to the other.
In the letter, Mr Swann told Mr Donnelly that his officials were facing "serious impediments" in monitoring travellers who cross the border having landed in the Irish Republic from an at-risk country.
Such passengers are required to self-isolate in Northern Ireland for 14 days.
Mr Swann said the system was reliant on people filling in a passenger locator form within 48 hours of entering Northern Ireland. He said if they failed to fill in those forms, the authorities here had no way of tracing them.
"The logical solution is to ensure that all international travellers landing on the island of Ireland provide data which is accessible for compliance checks in our respective jurisdictions," Mr Swann wrote.
"I would welcome the opportunity to engage with you on this with a view to finding a rapid resolution, including the introduction of new legislation and data-sharing agreements as required.
"I believe this would be a significant achievement which would demonstrate our commitment to working together to effectively address common challenges."
On Thursday, the Executive agreed to write to the UK and Irish governments requesting that a meeting of the British Irish Council is convened to discussing travel issues presented by Covid-19. The council includes the two governments and representatives from all the devolved regions in the UK.
There are some significant differences in policy adopted on both sides of the border. The Irish Republic's green list of countries deemed safe for travel only extends to 15 nations while a similar list issued by Stormont includes almost 60 destinations.
Ireland's Department of Health said consideration was being given to further enhancing north/south cooperation on overseas travellers, including adding information sharing.
"Minister Donnelly and minister Swann and their public health experts and officials are in close ongoing contact on a broad range of Covid-related issues," a spokesperon said.
"Contact tracing continues on an all-island basis, and information shared North and South, as it has been from the beginning of this public health emergency."