Kerry was due to meet later with Iranian Foreign Minister Mohammad Javad Zarif as negotiations on a nuclear deal move into a final phase.
While Kerry stressed that the main thrust of their talks in New York would be on the deal, and a looming June 30 deadline for a comprehensive treaty, he said he was "confident that Yemen will be mentioned certainly, because Iran is obviously a supportive party to the Huthis."
The UN Security Council also went into closed-door consultations on the crisis in Yemen and to hear former envoy Jamal Benomar give a final report.
The Moroccan diplomat resigned earlier this month after losing the support of Gulf countries for his mediation efforts as Shiite Huthi rebels pushed their offensive, amid accusations that they are being armed and supported by Iran.
Peace talks for Yemen collapsed after the Huthis captured the capital Sanaa and advanced on the southern city of Aden, forcing President Abedrabbo Mansour Hadi into exile.
"And I will certainly urge that everybody do their part to try to reduce the violence and allow the negotiations to begin."
The United Nations is working to re-launch peace talks but has run into hurdles over the venue for meetings, with Saudi Arabia insisting that the talks be held in Riyadh.
"Yemen's future should be decided by Yemenis," Kerry insisted, adding "all Yemenis have the right to come to the table," but he said the country's future "should be decided by Yemenis on both sides of the current dispute not by external parties and proxies."
But increasingly, US officials have acknowledged discussing other concerns with Iranian officials, even though the two countries do not have diplomatic relations.
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