They have reportedly sunk two ships and killed at least two crew members in that time.
They have claimed – often falsely – that they are targeting ships only linked to Israel, the US or the UK.
The Sounion was first hit by gunfire from two small boats last Wednesday, then struck by three unidentified projectiles, which sparked a fire and left it without engine power, the UK Maritime Trade Operations (UKMTO) office said.
Its 25 crew members were rescued by a European warship a day later and transported to Djibouti.
The oil tanker was later attacked again – the Houthis posted a video purportedly showing them setting it on fire.
A US State Department statement on Saturday, external expressed concern about the attacks on the Sounion. It warned of a possible spill of oil four times the size of the 1989 Exxon Valdez disaster, which released 257,000 barrels off the coast of Alaska.
On Tuesday Pentagon spokesman Maj-Gen Patrick Ryder said two tugs had been sent to salvage the stricken vessel but the Houthis had threatened to attack them as well.
He said the US was working with partners in the region to try to mitigate any potential environmental impact.
Read More: MV Sounion: US says tanker attacked by Houthis could be leaking oil