The US military did not confirm the Stryker-damage claim, but said that fighting had broken out overnight between fighters and Iraqi security units supported by US forces.
US forces carried out an air strike in Sadr City that killed "nine criminals", a US military statement said.
Nasser al-Rubaei, the head of al-Sadr's bloc in Iraq's parliament, told of "joint Iraqi and US attacks against Sadr City".
"We as the Sadr movement back the security forces," he said.
"But we reject that Iraqi forces be used as a shield for the occupation forces."
Sadr City conditions
In an effort to ease conditions for Sadr City's 2.5 million residents, the government has allowed lorries carrying maintenance teams, food, oil products and ambulances into the area.
A vehicle ban remains in effect as part of a curfew imposed on Baghdad after fighting broke out between government forces and Mahdi Army fighters on March 25.
The curfew has been lifted in the rest of Baghdad.
Al-Sadr has called for a protest on April 9 in Sadr City against the presence of US forces in Iraq.
His office said it expects at least one million people to turn out for the protest.
In northern Iraq, meanwhile, Iraq's security forces freed 42 university students who had been kidnapped by armed men on Sunday, a local army commander said.
Brigadier-General Khalif Abdul-Sattar said the students were waylaid about 30km south of Mosul on the main highway to Baghdad.
Three other students on a second bus were injured when armed men opened fire as the driver managed to speed away, he said.
It was not immediately clear where the students were coming from or where they were going.
Squeeze on al-Sadr
Sunday's clashes in Baghdad formed a violent backdrop to political efforts to isolated the Mahdi Army.
All the major Shia, Sunni and Kurdish parties have closed ranks to pressure al-Sadr into disbanding the Mahdi Army or be barred from political life, according to legislators and officials involved in the effort.
They said a first step would be to add language to a draft election bill banning parties that operate militias from fielding candidates in provincial balloting this autumn.
"We want the Sadrists to disband the Mahdi Army. Just freezing it is no longer acceptable," Sadiq al-Rikabi, a senior adviser to Nuri al-Maliki, the Iraqi prime minister, said.
Disarm or else
Members of the powerful Political Council of National Security met at the office of Jalal Talabani, Iraq's president, on Saturday and framed a 15-point statement aimed at disarming the militias, most of them aligned to political parties.
Hassan al-Rubaie, a Sadrist legislator, said on Sunday that "our political isolation was very clear and real during the meeting".
"We, the Sadrists, are in a predicament," he said.
"Even the blocs that had in the past supported us are now against us and we cannot stop them from taking action against us in parliament."
Al-Sadr controls 30 of the 275 parliament seats, a substantial figure but not enough to block legislation.