"Houthi leadership will find themselves sorely mistaken if they think this administration is going to let off the pressure, is going to let them off the hook for the reprehensible conduct that they continue to undertake," Price said in a press briefing.
Price added that the Biden administration will have more to say on the matter in the coming days.
Earlier on Wednesday, the Saudi-led military coalition operating in Yemen claimed that a civilian plane at Abha International Airport in southwestern Saudi Arabia caught fire as a result of an attack carried out by the armed Houthi political opposition faction.
Houthi military spokesman Yahya Saria responded by saying the opposition movement hit warplane airfields at the Abha airport via four drones on Wednesday afternoon and the strike "was accurate."
President Joe Biden has suspended US arms sales to Riyadh and reportedly informed Congress that the White House will remove the Houthis from the US foreign terrorist organization list as part of a US diplomatic effort to end the war in Yemen.
Yemen, the poorest country in the Middel East, has been engulfed in an armed conflict between the government forces led by the president-in-exile, Abd Rabbuh Mansur Hadi, and the armed Houthi political opposition faction since 2014. A coalition of countries led by Saudi Arabia in 2015 launched a military operation to support the internationally-recognized Hadi government. The Houthis continue to maintain control over a vast part of the north of the country, including the nation's capital of Sanaa.