CAIRO — The United Nations’ human rights chief warned on Tuesday that Yemen was on the brink of collapse, as health officials in the southern port city of Aden described a medical system failing after weeks of urban warfare that had left scores dead and hospitals overflowing with bodies.

The warning from the human rights chief, Zeid Ra’ad al-Hussein, came as a Saudi-led military offensive against the Houthis, a militia group from northern Yemen that Saudi officials have accused of serving as a proxy force for Iran, threatened to burst into a broader conflict.

The Houthis, acknowledging their alliance with Iran but denying acting on its orders, have been able to extend their offensive despite intensifying airstrikes by Saudi warplanes across Yemen.

There have been few signs that the battle, which began last Wednesday, is shifting decisively in favor of any of the combatants, raising fears of a lengthy war that is expanding the destabilizing regional conflict between the Persian Gulf monarchies and Iran. With Yemen under blockade from air and sea by the Saudi-led coalition, aid agencies intensified their warnings on Tuesday about the toll on civilians and hospitals, which are running critically low on medical supplies.