Yemen's Houthi 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-backed groups, have been harassing Red Sea shipping for months, triggering US and British reprisal attacks.

The Huthis have launched more than 30 attacks on commercial shipping and naval vessels since November 19, the Pentagon said on Tuesday.

The rebels say the attacks are in solidarity with the Palestinians and in protest at the Israel-Hamas war that has been raging in the Gaza Strip since October.

The attacks have prompted some shipping companies to detour around southern Africa to avoid the Red Sea, a vital route that normally carries about 12 percent of global maritime trade.

In response, US and British forces have launched two volleys of joint strikes targeting Huthi missile sites and other military facilities. The US has also carried out a series of unilateral strikes.

The US heads a coalition to protect Red Sea shipping, and is seeking to apply diplomatic and financial pressure by redesignating the Huthis as a "terrorist" group.