Normal conditions are not expected to resume until Monday following a country-wide ground stop, Duffy said, pushing back an earlier estimate of Sunday evening.
The head of the US Federal Aviation Administration (FAA), Bryan Bedford, had previously suggested that flights would be operating as normal by Sunday evening.
According to the flight-tracking portal FlightAware, by Sunday morning more than 1600 flights within, to or from the United States had already been cancelled, with an additional more than 3300 flights delayed.
On Saturday, nearly 6700 flights had been cancelled, and on Friday about 9200.
The FAA had halted all take-offs across the country on Friday in an unusual ground stop triggered by air traffic control communication issues.
The suspension lasted about four hours, leaving many travellers stranded.
Duffy said the technical problems have now been resolved but airlines need time to return aircraft and crews to their designated bases.
He previously said the disruption occurred during routine maintenance work carried out by the FAA's contractor L3Harris to replace old copper cables with fibre-optic technology in El Dorado, Arkansas.
In the process, a router had overloaded other devices in the air traffic control system with data requests, he said.
According to the transportation secretary, problems subsequently arose at a site in Amarillo, Texas, which had already been modernised weeks earlier.
Duffy said both routers are being inspected, adding that it remains unclear whether faulty equipment or installation errors triggered the disruptions.
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