The 3,000m (10,000ft) volcano is in Argentina's south-western Neuquen province, near the Chilean border. It began spewing ash and gas on Saturday morning.
In Argentina, the authorities have told residents of local villages to monitor the situation.
Many people have already left the area as a precaution.
Ash has been raining down on both sides of the border.
The Chilean authorities raised the alert - first issued on Saturday - to the highest level in Biobio region after the Copahue volcano increased seismic activity overnight.
Roads were being cleared for possible evacuation.
The plume of smoke is about 1,500m (5,000ft) high and the wind is pushing it away from Argentina, said Chile's emergency office (Onemi).
Planes flying over the southern Andes have also been warned to avoid the area.
Hundreds of flights were cancelled last year due to the eruption of Puyehue volcano, in Chile.
That eruption caused huge economic damage not only to property in the area but also to tourism in Bariloche and other resorts.
The Copahue eruption comes at the height of the tourism season.
Visitors from all over the world go to the area in the summer months to enjoy the mountains, lakes and also the Copahue and Caviahue thermal spas.
Copahue in the indigenous Mapuche language means "sulphur waters".

The 25-year-old lock progressed through the Pro12 club's academy system.
"It's my home region, where I've grown up, and it's an honour to be able to sign up for three more years," said the former Wales Under-20 international.
Ospreys head coach Steve Tandy said: "He doesn't get the credit he deserves outside of our environment, but he's consistent, he's diligent."

Police were called to the gorge, near Blair Atholl, at about 17:50.
Officers have been joined by the fire service, ambulance crews, a helimed and the local mountain rescue team for the search.
A spokesman for Police Scotland said: "There is currently no sign of the individual who is missing."
Chile has raised the alert level for the Copahue volcano to red, the highest possible.
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