The UN refugee agency UNHCR said the refugees, in four camps inside the border region, were in "desperate need" of help and currently out of reach.
"Concerns are growing by the hour," UNHCR spokesman Babar Baloch told reporters in Geneva.
Ethiopian Prime Minister Abiy Ahmed ordered troops into Tigray on November 4 following alleged attacks by Tigray People's Liberation Front forces on federal military camps in the northern region.
Abiy, who was awarded the Nobel Peace Prize last year, claimed victory on Saturday. Thousands have died so far and tens of thousands have fled into neighbouring Sudan.
UNHCR has had no humanitarian access to the refugee camps in Tigray since the start of the unrest.
It is thought that stocks that were delivered beforehand to the refugees, many of whom have fled Eritrea's authoritarian government, would not last beyond the start of this week.
"The camps will have now run out of food supplies, making hunger and malnutrition a real danger -- a warning we have been issuing since the conflict began," Baloch said.
"UNHCR appeals to the government of Ethiopia to continue to fulfil its responsibility in hosting and protecting Eritrean refugees and allow humanitarians to access people who are now desperately in need."
The four camps, which have been in place for more than a decade, shelter some 96,000 refugees.
Baloch also voiced alarm at unconfirmed reports of attacks, abductions and forced recruitment at the camps.
Meanwhile in Sudan, nearly 46,000 refugees from Ethiopia have now been registered, Baloch said, including more than 2,500 on Friday.
He said newly arrived refugees had reported seeing more checkpoints on roads from Ethiopia to Sudan, forcing them to take other routes.
UNHCR and its partners launched a plan at the weekend to provide urgent life-saving assistance including shelter, water and food, at a cost of $147 million, to meet the needs of up to 100,000 refugees over the next six months.