US tanks headed towards the city centre early on Wednesday, but were unable to advance after coming under rocket-propelled grenade fire launched by al-Mahdi fighters, reported Aljazeera's correspondent Atwar Bahjat.
US helicopters have also been hovering over the city, she added. Tanks backed by the helicopters pounded al-Mahdi fighters based in a cemetery, leaving four people dead, said hospital officials.
Another 29 people were injured in the attack. The casualty toll may be higher because it is difficult for ambulances to reach the area, said medics.
Fury still prevails in Najaf after a mortar bomb was fired at the shrine of Imam Ali, the first imam of the Shia, reported Aljazeera.
In other violence, two Russian contractors were killed in Iraq when their bus came under fire on Wednesday, reported the Foreign Ministry in Moscow.
Six alleged Iraqi resistance fighters were killed by US occupation forces in Tikrit, said the US military on Wednesday.
A US soldier was also wounded after apparently coming under fire late on Tuesday.
In a separate incident, a US soldier died and two others were injured in a road accident south of Tikrit on Tuesday night.
UN premier nominee
On the diplomatic front, the United Nations named Iraqi nuclear scientist Husayn al-Shahrastani as its leading candidate for prime minister in the country's interim government, reported The Washington Post on Wednesday.
Al-Akhdar al-Ibrahimi (R) has
met al-Shahristani several times
Al-Shahrastani, 62, has spent his years in exile focusing on humanitarian aid projects, and lacks any political affiliation, an asset that could allow him to serve as a bridge between Iraq's various factions, Iraqi officials were quoted as saying.
He has met several times this month with the UN envoy to Iraq whose interest in the nuclear scientist was piqued by a comment piece al-Shahrastani wrote last month for The Wall Street Journal, criticising the occupation authorities for not preparing Iraq for elections, said Iraqis familiar with al-Ibrahimi's mission.
Al-Shahrastani described himself as an adviser to Iraq's leading Shia cleric, Ayat Allah Ali al-Sistani. The Iranian-born cleric has forced the occupation to review their plans for Iraq several time.
US officials said negotiations for the 30-member caretaker government were still ongoing, but that al-Shahrastani has emerged by far as the most attractive candidate for prime minister.
The government will emerge after 30 June, the deadline for the so-called "transfer of sovereignty" from the occupation to Iraqis.
Al-Ibrahimi and US presidential envoy to Iraq Robert Blackwell are still trying to work out the "complicated geometry" of dividing power among Iraq's ethnic and religious factions, a senior US official in Baghdad told the daily.
After receiving a doctorate in nuclear chemistry at Canada's University of Toronto, al-Shahrastani served as chief scientific adviser to Iraq's atomic energy commission until 1979, when Saddam Hussein became president.
He was reportedly jailed for refusing to turn his attention from nuclear energy to nuclear weapons. He was held for 10 years at the notorious Abu Ghraib prison, from where he escaped in 1991.
Al-Shahrastani spent his initial exile with his wife and three children in Iran where he worked with Iraqi refugees. Later, he travelled to Britain where he was a visiting university professor.