The Ethiopian army has launched coordinated attacks on all fronts against forces from the northern Tigray region, the rebels told the BBC.
They said the government was using artillery, tanks, jets and drones in an attempt to “reinvade” the region.
A senior rebel source said Tigray forces were holding their ground.
The Ethiopian government has not confirmed any fighting and a communications blackout makes independent verification impossible.
A senior member of the Tigray People’s Liberation Front (TPLF), Getachew Reda, said the government offensive launched last week was now in “full swing”.
Asked if a ground offensive had been launched, Ethiopian Prime Minister Abiy Ahmed’s spokeswoman Billene Seyoum said the government had a responsibility to protect its citizens across the country from acts of terrorism, the Reuters news agency reports.