The people living in Ogaden are mostly nomadic
The ICRC has been carrying out water and sanitation projects there.
An army crackdown in the area after a series of rebel attacks has restricted the movement of essential goods.
The rebel group, the Ogaden National Liberation Movement, accuses the government of blockading the region, and producing a "man-made famine".
On Monday, the New York Times carried an article saying that Ethiopian troops were preventing emergency aid reaching the mainly Somali speaking region.
But aid agencies have been reluctant to complain publicly about the lack of access, fearing that it might compromise their work in the future.
The regional president of Ethiopia's Somali region, Abdullai Hassan, told the BBC that the ICRC had been given seven days to leave the area.
He accused the organisation of collaborating with the enemy and of spreading baseless accusations against the regional government on its website.
Ethiopia's eastern Ogaden region shares a long and porous border with Somalia, and most of its people are of the Somali ethnic group.
The ONLF has fought for the secession of the Ogaden region since the early 1990s.In April, rebels attacked a Chinese-run oil field killing nine Chinese and 65 Ethiopians.