KABUL, Afghanistan — Pakistan protested angrily Monday after the U.S.-led international force in Afghanistan confirmed that its helicopters staged cross-border air strikes last week against Pakistan-based Afghan militants “”in self-defense.””

Islamabad’s sharp reaction to the helicopter strikes Saturday came despite a long-standing understanding that allows the U.S.-led International Security Assistance Force to pursue militants who attack Afghanistan from bases in Pakistan’s rugged tribal territory.

ISAF usually informs the Pakistani military of any such incursion, but “”in this instance, there was no coordination until after, because of the imminent danger to the troops,”” said a U.S. defense official, who requested anonymity because of the sensitivity of the issue. “”This goes to the inherent right of self-defense.””

In a related development, U.S. Army Gen. David Petraeus, the top commander of coalition forces, said that U.S.-backed Afghan President Hamid Karzai had received overtures from senior Taliban leaders responding to his initiative to open peace negotiations.

“”There are very high-level Taliban leaders who have sought to reach out to the highest levels of the Afghan government, and they have done that,”” Petraeus told reporters after a tour of a detention facility for suspected insurgents at Bagram, the largest U.S. base in Afghanistan.

Afghan government officials, however, said the Taliban officials aren’t senior leaders.

The Pakistani protest appeared to be intended mostly for a domestic audience that deeply opposes U.S. attacks on insurgent strongholds on the Pakistani side of the border as violations of its country’s sovereignty.

The ISAF airstrikes Saturday were “”a clear violation and breach of the U.N. mandate under which ISAF operates,”” Pakistani Foreign Ministry spokesman Abdul Basit said in a statement. “”In the absence of immediate corrective measures, Pakistan will be constrained to consider response options.””

Pakistan lodged an official protest with ISAF, he said.

Basit didn’t elaborate on what “”corrective”” steps Pakistan was seeking or what responses it would consider if those steps weren’t taken.

The first cross-border air strike occurred after “”a significant number”” of insurgents launched an attack from Pakistani territory on a remote Afghan National Army base just inside Afghanistan’s eastern Khost province, ISAF said in its statement.

ISAF helicopter gunships monitoring the assault on Combat Outpost Narizah crossed into the North Waziristan area in pursuit of the militants, killing more than 30, ISAF said.

Additional helicopters arrived to assess the situation “”and received small arms fire again. The aircraft returned fire, resulting in additional insurgents killed,”” the statement said. “”At no time during the engagement did ground forces cross into Pakistani territory.””

ISAF said that the helicopters acted in conformity with the force’s “”rules of engagement.””

Abdul Hakim Ishaqzai, the Khost provincial police chief, said by telephone that between 60 and 70 militants died in the helicopter strikes.

Air Force Master Sergeant Jason Haag, an ISAF spokesman, said he couldn’t confirm reports in Pakistan that ISAF helicopters crossed the border again Monday, attacking militants in the tribal agency of Kurram.

“”There was an operation near the border, but I don’t have information right now that tells me they crossed the border,”” he said, giving the location of the operation as the eastern Afghan province of Paktiya.

The United States has been stepping up air strikes on extremist targets in Pakistan’s tribal area, using unmanned missile-firing drones operated mostly by the CIA. On Monday, a missile attack on a house in a village in the North Waziristan tribal area killed at least two militants, according to news reports in Pakistan.

ISAF, which works closely with the Pakistani military, rarely reports cross-border attacks by ISAF helicopters or jetfighters.

“”This should be considered a watershed event,”” said Mehmood Shah, an analyst who served as the top security official for Pakistan’s tribal area. “”Our units should be deployed to fire upon them. This border has sanctity. NATO must realize they have a mandate to operate in Afghanistan, not in Pakistan.””

The helicopter attacks could increase the pressure on Pakistan’s governing coalition, which is deeply unpopular because of its close U.S. ties, widespread corruption and its slow, inadequate response to the country’s recent devastating floods.

Marvi Memon, an opposition Parliament member, said she plans to raise the issue in the legislature.

“”Self-defense is no excuse for violating Pakistani airspace and thus, our sovereignty,”” she said.

Afghan officials have long accused Islamabad of allowing al-Qaida-linked Afghan insurgent groups to maintain sanctuaries inside the tribal areas as part of a plan to replace the U.S.-backed government in Kabul with a pro-Pakistan administration.

Islamabad rejects the charge, and points to military operations that it has undertaken against militant sanctuaries, although those offensives have targeted Pakistani extremists, not Afghan insurgents.

U.S. commanders privately complain that Pakistani security forces often do nothing to prevent insurgents from launching attacks on Afghan territory from the tribal area or from infiltrating Afghan territory.

Petraeus’ comment was the most explicit to date by a senior U.S. official suggesting that contacts are taking place between Karzai and top Taliban leaders, who are thought to be based in Pakistan. Kabul previously has acknowledged discussions with a separate militant group, Hezb Islami.

“”Now President Karzai’s conditions are very clear, very established, and, certainly, we support them as we did in Iraq, as the British did in Northern Ireland,”” he said. “”This is the way you end insurgencies.””

Karzai has said he is prepared to make peace with insurgents who renounce violence, accept the Afghan Constitution and reject al-Qaida.

Waheed Omar, a spokesman for Karzai, said that contacts “”at different levels of the Taliban”” had been going on “”for weeks, if not months.”” However, he said those now in contact can’t be characterized as the movement’s senior leaders.