US aircraft attacked the Old City of Najaf late on Monday, bombing suspected Shia militant positions.
The latest attacks came after dark
Several large explosions were reported as an AC-130 plane circled overhead, and gun battles broke out nearby.
Earlier in the day correspondents said plumes of black smoke appeared near the sacred Imam Ali Shrine as US troops appeared to be closing in.
US and Iraqi efforts to persuade Shia cleric Moqtada Sadr to give up the holy site have so far failed.
US tanks and snipers took up positions near the mosque on Monday night, stepping up the pressure on Mr Sadr and his followers to end their armed resistance.
The operation followed US assaults on targets the previous night, and continued fighting during the day.
The BBC's Alistair Leithead went into the shrine, and reported up to 1,000 apparently unarmed supporters of Mr Sadr inside.
He saw two men who had been wounded by gunfire carried into the shrine, amid American sniper fire.
A spokesman for Mr Sadr said he was still hopeful of a deal, but a ceasefire would only happen if the Americans went back to their bases. If that happened, he said, Mr Sadr's Mehdi Army militia would disappear back into the community.
In other Iraq developments:
US aircraft and artillery hit targets in Falluja, west of Baghdad, which they claim were hiding fighters loyal to Jordanian militant Abu Musab al-Zarqawi
officials in the Baghdad Shia slum district of Sadr City said four Iraqis were killed in clashes or a bomb blast
The price of oil drops slightly as Iraqi crude oil exports are said to be back to normal after several disruptions
A Turkish contractor and
two Iraqis have been killed in an ambush in the northern city of
Tikrit, a US military spokesman says
French-American journalist Micah Garen is released by his kidnappers in the southern city of Nasiriya
Meanwhile, the Iraqi health ministry has announced that 114 fighters and civilians were killed in Najaf between noon on Saturday and noon on Sunday.
Tawfiq Mohammed of Najaf General Hospital said at least three further people were killed and 18 injured during Sunday night's fighting.
The confrontation is now in its third week. Iraqi government officials say they intend to resolve the crisis without raiding the Imam Ali
shrine, one of Shia Islam's holiest sites.
Senior government officials said last week that an Iraqi force
was preparing to raid the shrine within hours to expel the
militants, but correspondents say such an operation would anger Shias across
the country and could turn them against the new interim government.
Mr Sadr's whereabouts remain unclear - his aides denied claims by Iraqi police that he had fled the city and was heading north.