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.
The lakes, mountains and thermal spas in the area are popular with tourists from around the world 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".
Tungurahua means 'throat of fire' in the local indigenous language People living on the slopes of the Tungurahua volcano in Ecuador are being evacuated as a plume of gas and ash rises from its crater.