Asda CEO Andy Bond said that although there were no plans to open stores in the Republic, "where we can do home shopping, we will".
"Northern Ireland stores are particularly strong at the moment and cross-border trade is phenomenal," he said. "Christmas sales per square foot were unbelievable and about half the sales were in euros. We can investigate whether we can bring home shopping to southern Ireland."
First, however, the retailer would have to overcome difficulties delivering to addresses in the Republic, which does not use UK-style postcodes, said Joe Shields at the Centre for Cross Border Studies. While Dublin has a system of area codes, in rural parts of the country the postal service uses the full address to make a delivery.
The UK satellite system used by delivery drivers would be of no use and the Asda website would have to be adapted.
Asda is understood to be working on a solution, but distribution companies are already trying to take advantage. Peter Beckett, manager of Asda's store in Enniskillen, said he had been approached by several touting for business. One told him it had 500 customers in the Republic who would pay a £20 delivery charge to get Asda groceries delivered.
Meanwhile, the level of cross border traffic continues to rise. At Sainsbury's Newry store, 40% to 50% of transactions were in euros on some days, said deputy store manager, Andy Scanlon. "We get people travelling right from the bottom of Ireland."
Last month, the Republic's minister for finance, Brian Lenihan, branded those who shopped in Northern Ireland unpatriotic, Superquinn announced it was closing its Dundalk store and axing 400 jobs and Tesco said it was cutting workers's hours in five stores in the Republic.