Yemen's Huthi rebels fired "several" missiles at a US warship in the Red Sea, they said on Wednesday, hours after the US military reported shooting down a missile.

"The naval forces of the Yemeni Armed Forces, with the help of God Almighty, fired several... naval missiles at the American destroyer USS Gravely in the Red Sea," the Huthi statement said.

Earlier, US Central Command said the USS Gravely shot down a Huthi missile at about 11:30 pm (2030 GMT) on Tuesday -- the latest in a series of such incidents in the Red Sea.

"Iranian-backed Huthi militants fired one anti-ship cruise missile from Huthi-controlled areas of Yemen toward the Red Sea," a Centcom statement said.

"The missile was shot down by USS Gravely. There were no injuries or damage reported."

The Huthis, part of the anti-Western, anti-Israel "axis of resistance" of Iranian loyalist groups, have been harassing Red Sea shipping for months, triggering US and UK reprisal attacks.