A spokesman for the militant group said Monday they had already hit one such cargo ship.
That came after a U.S. military spokesman said earlier that a U.S.-owned freighter had been struck by a missile.
No injuries or significant damage were reported.
The Houthis have been targeting vessels in the region since November, focusing on ones they say have a connection to Israel.
They say that's in solidarity with Palestinian Islamists Hamas over the conflict in Gaza.
The Houthis say U.S. and British ships have become "legitimate targets" following strikes on Yemen by the two countries last week.
Now there are signs that the crisis is causing mounting trouble for shipping, and all the firms that rely on it.
On Monday, more oil tankers altered course or paused their journeys to avoid the region.
That threatens to delay and disrupt crude shipments, though the impact on prices has so far been modest.
On Tuesday, oil traded slightly lower, as traders balanced the Red Sea concerns against worries that weak global demand could sap prices.
Some tankers carrying Qatari natural gas towards the Suez Canal also paused their journeys, though they appeared to be back on course by Tuesday.
The impact on container shipping has been more marked, with many vessels moved to safer but much longer routes.
Carmakers Tesla, Suzuki and Volvo have already warned that they will have to curb some output as a result, due to parts not arriving in time.
