"The enemy, having already relocated a massive military force to Eritrea, has now begun a joint campaign with the foreign invading force of Eritrea to brutalise and exterminate the people of Tigray," the rebels' military command said in a statement.
AFP was not able to independently verify the claims. Access to northern Ethiopia is severely restricted and Tigray has been under a communications blackout for over a year.
TPLF spokesman Kindeya Gebrehiwot told AFP the attack was coming "from Eritrea".
Another TPLF spokesman, Getachew Reda, said on Twitter that the rebels were "defending their positions" and reported "heavy shelling" from some locations.
Addis Ababa has not responded to requests for comment about the reported offensive.
Fighting between government forces and the rebels resumed last week after a five-month lull.
Clashes on the ground and air raids over Tigray have dashed hopes of peacefully resolving the nearly two-year war.
Combat had been concentrated around the southeastern border of Tigray, with the rebels pushing into the neighboring Amhara and Afar regions, sending residents fleeing.
The government on Wednesday accused the TPLF of launching a wider "invasion" into parts of western Tigray, and other areas west of the initial clashes.
The rebels in turn alleged that the government and its neighbour Eritrea -- which backed federal forces during the war's early phase -- were responsible for opening a new front.
On Thursday, the government said TPLF attacks "had further continued to intensify".
"Innocent civilians are being killed; many are being displaced and property is being destroyed," the Government Communication Service said in a statement.
Fighting has spread since fighting erupted over a week ago, while the capital of the war-torn Tigray region has been hit twice by air strikes.
The first air raid on Tigray's capital Mekele killed at least four people, including children, in a strike UNICEF said "hit a kindergarten".
The second, around midnight Tuesday, caused injuries and property damage, the TPLF said.
Both sides accused each other of firing first and shattering a March truce that had paused the worst of the bloodshed in northern Ethiopia.
The renewed fighting has alarmed the international community, with UN chief Antonio Guterres and senior diplomats from the European Union, Britain and the African Union (AU) among those to appeal for restraint.
US Secretary of State Antony Blinken on Thursday called on the Ethiopian government and the TPLF "to immediately halt military operations and redouble efforts to bring a permanent end to the conflict".
"We remain deeply concerned at the resumption of fighting and the lives that it puts at risk," he said on Twitter.
Legesse Tulu, a spokesman for the Ethiopian government, responded on Twitter, saying: "The US Government needs to cease the 'both parties' rhetoric and denounce the destructive path of TPLF affecting various Ethiopian communities."
The government said Thursday that the TPLF was diverting aid supplies to its fighters.
US aid chief Samantha Power said humanitarian workers in Tigray, Afar and Amhara were "facing unacceptable interference -- including detentions, looting of fuel, and vehicle seizures".
"USAID reiterates our call for all parties to respect humanitarian operations so that aid can reach those in need," Power posted on Twitter on Wednesday.
