“This is a moment to be seized and built on and a real opportunity to start an inclusive political process under UN auspices to sustainably end the conflict,” Mr Grundberg told AP.
Yemen's Foreign Minister Ahmad Awad bin Mubarak said Saudi-Houthi talks never stopped, even during the war.
“Talks between the Saudis and the Houthis never stopped even during the war, whether on political or security issues,” he said in an interview with Cairo TV on Sunday.
A senior Saudi official confirmed this at an event attended by The National.
On Monday, Iran's Foreign Ministry said it supports a political peace in Yemen.
Iran has long been accused of supporting the Houthis through weapons smuggling and providing technical know-how, accusations detailed in UN reports between 2019 and this year, and the US and British navies have intercepted arms shipments.
Last month, Saudi Arabia and Iran announced the restoration of diplomatic ties under a China-brokered deal that will involve the resumption of direct flights between the two countries and the reopening of embassies for the first time since 2016.
A delegation from Saudi Arabia visited Tehran at the weekend to discuss reopening its embassy. Iran said it will send a delegation from Tehran to Riyadh to do the same.
Yemen's Information Minister Moammar Eryani lauded the Saudi role in Yemen.
“Progress for achieving political peace in Yemen is a win for constitutional legitimacy and the coalition led by our brothers in the Kingdom of Saudi Arabia … especially in light of the positive developments in the region of which the Saudi-Iranian agreement is the most important,” he wrote in a Tweet on Monday.
“The time is ripe more than ever before to achieve peace.”