"We call on President Saleh not to seek to wait out the situation, and to seriously consider and accept the mediation offer made by the Gulf Cooperation Council," Peschke said.
UN Secretary General Ban Ki-moon said he was "deeply troubled" by the violence and and called on all sides to find a peaceful solution.
Ban "is deeply troubled by the violent clashes in the centre of Sanaa between the security forces of the government and armed tribesmen that have left many people dead and wounded," his spokesman Martin Nesirky said.
Nesirky said Ban "is concerned that this confrontation might further destabilise the situation and calls for an immediate end to the fighting".
Heavy clashes between dissident tribesmen and elite Republican Guard troops triggered the closure of Sanaa airport, aviation and tribal sources said.
The clashes pitted clansmen of the Arhab tribe of hardline cleric Abdul Majid al-Zindani, who faces US sanctions as a "terrorism financier," against Republican Guard troops under the command of Saleh's son Ahmed.
The ancestral homeland of Al-Qaeda leader Osama bin Laden, Yemen has been seen as a key partner in the US "war on terror" but in recent days Washington has stepped up its pressure for a negotiated depature from office for Saleh, in power since 1978.
Fighting also raged between security forces and armed civilians loyal to dissident tribal chief Sheikh Sadiq al-Ahmar despite mediation efforts by impoverished Yemen's wealthy Gulf Arab neighbours.
Sheikh al-Ahmar, who heads the powerful Hashid tribal federation, fled from his home in the Al-Hasaba neighbourhood of north Sanaa after it came under missile attack, tribal sources said.
Three days of fighting have killed more than 44 people in the Yemeni capital, according to an AFP tally based on reports by medics, the government and tribesmen.
As the fighting raged, armed tribesmen from Amran, 50 kilometres (30 miles) north of Sanaa, headed towards the capital to join the battle against Saleh's forces, a tribal source told AFP.
Tribesmen had occupied the offices of the state news agency Saba and national airline Yemenia. They also tried to storm the interior ministry headquarters, witnesses and a high-ranking Yemeni official said.
The latest clashes came despite an appeal by Saleh late on Tuesday for supporters of the dissident tribal leader to "cease their aggression on security forces."
Clashes between security forces and Sheikh al-Ahmar's followers broke out on Monday after Saleh refused to sign a deal with the opposition sponsored by the six-nation Gulf Cooperation Council that would see him leave office within 30 days in return for a promise of immunity from prosecution.
The sheikh pledged his support in March to protesters who have been seeking to topple Saleh since late January.
One of the 10 sons of Sheikh Abdullah al-Ahmar, who was Saleh's main ally until his death, Ahmar is capable of rallying thousands of armed supporters, tribal sources say.
Tribal loyalties run deep in Yemen, the Arab world's poorest country, which has an estimated 60 million firearms in private hands, roughly three for every citizen.