Jamie McGoldrick told reporters at United Nations headquarters in a phone call from Sanaa that there had been no airstrikes or other military activity and "an intense calm has returned to Sanaa in the last day or so".
Death toll from week-long clashes in the Yemeni capital Sanaa has increased to 234 people, while 400 others were wounded, including 383 severely injured, the International Committee of the Red Cross (ICRC) for the Middle East said on Tuesday.
"This must be done for humanitarian reasons immediately", the president added.
"We also have taken note of the reported killing earlier today of former President Ali Abdullah Saleh as well as several of his associates", he said.
The Saudi coalition appears eager to facilitate this, with UAE Crown Prince Mohammed bin Zayed al-Nahayan visiting Saleh's exiled son to offer condolences, and raising speculations that the younger Saleh could soon be sent to Yemen to try to lead what's left of the Saleh force.
He said over 22 million Yemenis need humanitarian assistance and over 8 million are on the brink of starvation.
Saudi Arabia and its allies intervened in neighbouring Yemen in March 2015 to push back the rebels who control the capital Sanaa, in an attempt to restore the government of Abd Rabbo Mansour Hadi to power.
The clashes then intensified and turned into street fighting after Saleh on Saturday announced disengagement of his alliance with the Houthis and declared war against them, citing allegedly "Houthi assaults on his family members".
Saleh ruled Yemen for more than three decades until an uprising forced him to step down in 2012. That helped propel Yemen into the ruinous civil war that has spread hunger and disease among its 28 million people. The port of Hodeida is the main conduit for UN-supervised deliveries of food and medicine to Yemen, where poverty has been compounded by war and a blockade on ports and airports imposed by the Saudi-led coalition.