"I take this opportunity to urge all states with influence in the region to encourage, by all possible means, the resolution of this crisis," Turk told a special session of the Human Rights Council in Geneva.
"I strongly condemn this wanton violence, in which both sides have trampled international humanitarian law."
At the Geneva meeting, Britain's minister of state for development and Africa Andrew Mitchell urged the 47-member council to "send a united message of concern and horror" ahead of an expected vote.
Meanwhile, Sudan's ambassador told the council: "What's happening in Sudan is an internal affair."
Clashes rocked Halfaya, an entry point to the capital, early on Thursday as residents heard warplanes circling over Khartoum and its adjoining sister cities of Bahri and Omdurman, but the fighting appeared calmer than on Wednesday.
Talks on Sudan's crisis continued Wednesday in Jeddah, sponsored by Saudi Arabia and the United States.
After days of no apparent movement, a mediation source told Reuters on Wednesday that the negotiations had made progress and a ceasefire agreement was expected soon.
Talks continued late into the night.
Army general Yassir al-Atta was quoted on Thursday saying the talks should aim at removing the Rapid Support Forces (RSF) from Khartoum, merging its fighters into the regular military and putting its leaders on trial.
"Any dialogue outside those points is simply delaying the war to another time," he told Asharq al-Awsat newspaper, adding the army had beaten back RSF forces at one key Khartoum location.
U.S. Under Secretary of State Victoria Nuland on Wednesday said U.S. negotiators were "cautiously optimistic" about securing a commitment to humanitarian principles and a ceasefire but were also looking at appropriate targets for sanctions if the warring factions did not back this.
