The military says a Multi-National Division - Baghdad soldier died in an explosion Sunday during a route-clearing patrol in an eastern section of the capital.
A separate statement says another American soldier also has died of wounds suffered in a roadside bombing north of Baghdad.
That raises to two the number of Multi-National Division - North soldiers killed in the blast in the volatile Diyala province. The other death was reported Sunday after the attack occurred.
At least 4,020 members of the U.S. military have died since the Iraq war started in March 2003. That's according to an Associated Press count.
Meanwhile, Iraqi security forces said a large explosion demolished a building in the southern city of Basra on Monday, killing at least three militants and wounding four.
British military spokesman Maj. Tom Holloway said it was not clear what caused the blast but insisted no British, U.S. or Iraqi forces were involved. An Iraqi police official and witnesses say those killed were gunmen using the abandoned house as a base, and that the building was destroyed in an air strike.
The official spoke on condition of anonymity because he was not authorized to release the information.
U.S.-led forces have carried out several airstrikes amid fierce clashes between Shiite militiamen and Iraqi troops. But Holloway says that was not the case this time.
On Sunday, suspected Shiite militants lobbed rockets and mortar shells into the U.S.-protected Green Zone, killing two American troops and wounding dozens more, officials said.
The attacks occurred as U.S. and Iraqi forces battled Shiite militants in Sadr City in some of the fiercest fighting since radical cleric Muqtada al-Sadr ordered a cease-fire a week ago. At least 16 Iraqi civilians were killed and nearly 100 wounded in the fighting, according to hospital officials.
Prime Minister Nouri al-Maliki issued his strongest warning to date on Monday to al-Sadr to disband his militia or face political isolation. The Sadrists said Monday a move to ban them from elections would be unconstitutional.
Aide Hassan al-Zarqani said from Iran that al-Sadr will consult Grand Ayatollah Ali al-Sistani and other top Shiite clerics if the government continues to pressure al-Sadr to disband the militia or see his candidates banned from upcoming elections.
A military official said two U.S. troops died and 17 were wounded in the attack on the Green Zone, which houses the U.S. Embassy and the Iraqi government headquarters in central Baghdad.
A senior U.S. military official, also declining to be identified for the same reason, said the rockets were fired at the Green Zone from Sadr City while the mortar shells came from another predominantly Shiite neighborhood in eastern Baghdad, New Baghdad.
U.S. commanders have blamed what they call Iranian-backed rogue militia groups for launching missiles against American forces. American commanders say Iran supplies the militias with armor-piercing roadside bombs known as explosively formed penetrators or EFPs and other weapons. Tehran denies the allegations.
In related developments:
The deadly mortar and rocket strikes in Baghdad came despite a strong push by the U.S. military to prevent militants from using suspected launching sites on the southern edge of Sadr City, the Baghdad stronghold of the Mahdi Army of anti-U.S. cleric Muqtada al-Sadr.
Fierce fighting erupted in Sadr City earlier Sunday after Iraqi troops backed by U.S. soldiers and attack helicopters tried to advance deeper into the enclave of some 2.5 million people.
American helicopters also fired Hellfire missiles that destroyed a vehicle and killed nine militants who were attacking Iraqi security forces rocket-propelled grenades in the area, the military said in a statement.
The surge in violence came as tensions rose in Shiite areas despite al-Sadr's cease-fire order issued March 30 that ended nearly a week of clashes in Baghdad, Basra and other cities in the Shiite south.
The cleric stopped short of asking his fighters to surrender their weapons, and sporadic clashes have continued.
The inability of the Iraqi security forces to curb the militias has cast doubt on their ability to take over their own security two days before the top American officials in Iraq - Gen. David Petraeus and Ambassador Ryan Crocker - are to brief Congress on the prospects for further reductions in the U.S. troop presence in Iraq.
Al-Sadr has called for a "million-strong" anti-U.S. demonstration on Wednesday in Baghdad to protest the fifth anniversary of the capture of Baghdad by invading U.S. troops.
CBS News chief foreign correspondent Lara Logan reports that any action against the Mahdi Army will likely bring more violence, and the government's ultimatum comes at a crucial point this time - coinciding with al-Sadr's call for a march.
At the edge of Sadr City, Lt. Col. Dan Barnett, the commander of the 1st Squadron, 2nd Stryker Cavalry Regiment, said Iraqi forces had come under sustained fire overnight after establishing checkpoints deeper into the Shiite district.
"They're working to establish control," he said, speaking to a small group of reporters as heavy gunfire resounded outside a joint U.S.-Iraqi base.
The Iraqi government has relaxed security measures Saturday around the Mahdi Army strongholds of Sadr City and the Shula neighborhood, allowing trucks carrying maintenance teams, food, oil products and ambulances into the areas that still face a vehicle ban despite the lifting of a citywide curfew.
But residents continued to complain of hardships.
"Our situation is miserable. We lack food, water and electricity. This morning I saw two men being shot by a sniper as they were trying to cross the street near my house. The government should do something to end our suffering," said Hussein Khazim, a taxi driver who has been out of work since the turmoil erupted in late March.