Eliasson made the appeal when he was speaking to the press in Geneva after his travels over the past several days to Saudi Arabia, the United Arab Emirates (UAE), Iran, and Turkey, Haq said at a daily news briefing here.
"Asked about local ceasefires in Syria, the deputy secretary-general said that the United Nations would very much encourage every attempt to create local ceasefires," Haq said.
"In the absence of stopping the fighting all over the country, he said, we should at least now, before the winter starts, try to de-escalate and reduce the level of violence."
Eliasson said that "he had discussed the wars in Syria and Yemen as well as the situation in the Middle East region during his visits to those countries," the deputy spokesman said.
Syria and Yemen were among the hectic topics during the annual high-level debate of the UN General Assembly, a week-long event held here late last month.
"Regarding Yemen, the deputy secretary-general said that he detected interest from Saudi Arabia as well as the UAE and Iran to encourage talks to start as soon as possible," Haq said.
"He noted that his special envoy for Yemen, Ismail Ould Cheikh Ahmed, is proposing talks within the next few weeks and added that we very much hope that these talks will take place," he added.
On Syria, he said, the officials he met with had discussed Russia's actions in Syria, and there was a hope expressed, particularly from the Saudis and the UAE, that that military action would be solely directed towards terrorism coming from the Islamic State.