Taliban militants staged an assault against high-profile coalition targets in central Kabul Tuesday and at least two other attacks in another part of town — a series of brazen strikes by the tenacious Islamist movement in Afghanistan’s capital.

“Our insurgents attacked in Kabul city,” Taliban spokesman Zabiullah Mujahid told CNN. “Their target is the U.S. Embassy, governmental organizations and other foreign organizations.”

A coalition officer and a senior ISAF official both confirmed to CNN that there had been intelligence that insurgents might launch a high-profile attack in the capital around the 10th anniversary of the Sept. 11, 2001 attacks.

Militants opened fire near the U.S. Embassy and NATO’s International Security Assistance Force headquarters after they stormed into a nearby abandoned building, U.S., NATO and Afghan officials said.

ISAF said the insurgents attacked the “vicinity” of the sites using small arms and rocket-propelled grenades. It said Afghan National Security Forces and coalition forces immediately responded and were at the scene, and coalition forces were providing air support.

A senior ISAF official said there are less than 10 insurgents involved in the attack, launched from an area outside the secure zone surrounding both areas.

Sidiq Sidiqi, an Interior Ministry spokesman, said a police officer was killed and security forces killed “two of the suicide attackers in the building.” He said police think there are two or three more attackers in the structure.

The abandoned building is next to Abdul Haq Square, a few blocks from the U.S. Embassy, an intelligence officer told CNN.

An embassy spokeswoman said staff at the U.S. facility have been told to take cover and there were no casualties at present among embassy personnel.

“The U.S. Embassy can confirm an attack has occurred in the area of the U.S. Embassy” by people armed with rocket-propelled grenades and small arms, spokeswoman Kerri Hannan said.

ISAF personnel currently are being told to stay indoors and take cover but there are no casualties, the official said. Gen. John Allen, ISAF commander, remains on the compound, ISAF said.

The Afghan National Directorate of Security and the Interior Ministry spokesman said a firefight with insurgents was ongoing. Militants were using rocket-propelled grenades and light weapons. A hospital official said two injured civilians were being treated and the hospital expects to handle more casualties.

Central Kabul is considered a high-security area, and it is protected by police and other security forces.

The area has been cordoned off, according to Gen. Mohammed Zahir, the head of criminal investigation for the Afghan police.

The coalition and ISAF officials pointed out that while this attack is considered to be high-profile because of the U.S. embassy and ISAF targets, it so far is not considered a “spectacular” attack because of the small number of gunmen and limited weapons involved.

The Taliban said a suicide attack also targeted a police compound in the western part of the city. Two police officers were injured, the Interior Ministry said. A suicide attacker clad in an explosives vest was shot before he could enter the compound.

A suicide attack injured a policeman and a civilian in front of a high school in the western part of the city.

The U.S. Embassy issued an emergency message about the series of attacks across Kabul on Tuesday, saying “the situation is uncertain and ongoing. There are media reports that many roads are closed in Kabul.” It said appointments for visas or U.S. citizen services have been canceled for now and it said Americans in Afghanistan should monitor the websites of the embassy and the State Department’s Bureau of Consular Affairs for the latest information.

“We urge U.S. citizens to shelter in place, avoid unnecessary movement, and to avoid the neighborhood around the U.S. Embassy: Wazir Akbar Khan, Microrayon, and Massoud Circle,” it said.

“The Embassy also urges U.S. citizens to remain vigilant and avoid areas where Westerners congregate. Do not discuss travel plans or other personal matters with strangers, or in public. Be alert and aware of your surroundings, and always travel with mobile phones or appropriate communication equipment.”

These are the latest attacks by the Taliban in a violent period. Over the weekend, two Afghan laborers were killed, and 77 U.S. troops and 25 Afghan workers were wounded in a Taliban truck bombing on an ISAF base in Afghanistan’s Wardak province.

Last month, Taliban suicide bombers killed at least eight people in Kabul when they attacked the British Council, a British-government-affiliated body that fosters cultural and academic exchanges.

The Taliban harbored the al-Qaida network that launched the 9/11 strike. A U.S.-led coalition toppled the Taliban and has been fighting militants in Afghanistan for nearly 10 years.

NATO is drawing down and handing over security control to national forces. Some 10,000 U.S. troops are scheduled to depart by year’s end, with all U.S. military personnel out of Afghanistan by the end of 2014.

NATO Secretary-General Anders Fogh Rasmussen said the Taliban attacks in Kabul Tuesday were an attempt to derail plans to hand over security responsibility to Afghan forces in Afghanistan.

“We are witnessing that the Taliban try to test [the] transition but they can’t stop it. Transition is on track and it will continue,” Rasmussen said.

CNN’s Joe Sterling in Atlanta, Barbara Starr and Elise Labott in Washington and Tim Schwarz and Ruhullah Khaplakwak in Kabul contributed to this report.