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 but gave no further details.
The 11-month conflict has caused a humanitarian crisis, with the United Nations warning that about 400,000 people were living in famine-like conditions in Tigray in July.
Thousands of people have been killed in the conflict, and another two million have been forced to flee their homes.
Difficult to justify an offensive
Analysis by Kalkidan Yibeltal, BBC News, Addis Ababa
The Ethiopian government has not confirmed reports of a major offensive against the Tigray rebels, but a mobilisation of the country's forces and allied militia over the past few months left no doubt that a huge operation was in the offing.
With elections out of the way and the rainy season at its end, Prime Minister Abiy seems focussed on a military solution to the long-drawn out conflict.
The war has already taken its toll on the country: thousands have been killed, millions displaced and hundreds of thousands are facing famine.
But both sides have sent mixed signals about their willingness to accept a peaceful resolution.
It will, however, be a difficult task for Mr Abiy to justify another major offensive at a time when his government is facing a lot of international pressure to open talks with the rebels.
The Ethiopian army took control of most of Tigray in November 2020, after TPLF forces seized a military base.
In June 2021, the rebels recaptured Tigray in a surprise attack, and then moved into parts of neighbouring regions like Amhara.
Ethiopia has declared the TPLF a terrorist organisation, but the TPLF insists that it is the legitimate government in Tigray.
Tigray - the basics
- Since 1994, Ethiopia has been divided into states, now numbering 10; they are defined on ethnic grounds by the constitution and described as largely autonomous, but with central institutions
- In 2018, following anti-government protests, Abiy Ahmed took over as prime minister and introduced reforms
- Powerful politicians from Tigray, Ethiopia's northernmost state, accused Mr Abiy of trying to increase federal power
- Relations worsened and, after the government accused Tigrayan rebels of attacking military bases, the Ethiopian army moved in in November, backed by Eritrean troops
- Mr Abiy declared the conflict over in late November, but fighting has continued
Mr Getachew said the current offensive was taking place in towns in Amhara region, and also involved Amharan regional forces fighting alongside the army.
The Tigray forces' statement says "hundreds of thousands of regular and irregular fighters" have been fielded in the current assault.