The refugees have not received any assistance for weeks, Grandi said.
Human rights abuses have been reported, including killings, intentional abductions and the forced return of refugees to Eritrea.
Satellite images show fires and other destruction in the camps.
"All this clearly indicates serious violations of international law," RIA Novosti quoted Grandi as saying.
In December, it was reported that the number of Ethiopian refugees who arrived in Sudan to escape the fighting in Tigray state reached almost 46,000.