"This breaks our heart," Gates told a news conference at the presidential palace in Kabul, as Karzai looked on.

"Not only is their loss a tragedy for their families, it is a setback for our relationship with the Afghan people, whose security is our chief concern."

The comments on the first day of a two-day trip to Afghanistan came after Karzai angrily rejected a public apology from General David Petraeus, the US commander of international troops in Afghanistan, over the incident.

US President Barack Obama has also voiced "deep regret."

Nine young boys died in the NATO air strike targeting insurgents last week, as the children collected firewood in the eastern province of Kunar.

In response to Gates, Karzai said he respected the apology but was in a difficult position over civilian casualties and called for them to be halted altogether.

Civilian deaths from coalition military operations against Taliban insurgents have long been a source of friction between Kabul and its Western allies.

But Karzai's rejection of Petraeus's apology as "not enough" came at a particularly sensitive time for the military effort in Afghanistan, where 97,000 of the roughly 140,000 foreign troops are from the United States.

Afghan army and police are due to take control of security across the country in 2014, allowing foreign forces to leave combat roles, with the transition process starting from July this year.

Highlighting Afghan anger over the issue, hundreds of people took to the streets of Kabul on Sunday to protest against the deaths, shouting slogans including "Death to America -- death to the invaders".

Gates said NATO-led forces had been working "extremely hard" to avoid civilian casualties and had made progress in the past two years.

But he added that he understood how "these tragedies weigh heavily" on Karzai and "create problems for him as the leader and the protector of the Afghan people."

After issuing his apology, the Pentagon chief expressed optimism that US and allied troops will be ready to start a limited withdrawal from Afghanistan in July, as promised by Obama.

He suggested only a small number of troops may be pulled out while others may be shifted to other areas as Afghan forces expand.

"While no decisions on numbers have been made, in my view we will be well-positioned to begin drawing down some US and coalition forces this July, even as we redeploy others to different areas of the country," he said.

Karzai is expected to give details on March 21 of where the transition process will start.

In comments before American troops earlier at Bagram airfield, Gates stressed the United States was ready to offer military assistance even after the planned handover of security to Afghan forces.

"We are fully prepared to have a continuing presence here assisting the Afghans after 2014," he said. "Obviously it would be a small fraction of the presence that we have today."

The Taliban and other militants have been waging a bloody insurgency against international and Afghan government forces since being ousted by a US-led invasion in 2001.

Press secretary Geoff Morrell told reporters travelling with Gates that while this was "not a decision-making trip" on future troop levels, it would "certainly inform him (Gates) on making those decisions in coming months."

The defense secretary, who plans to step down this year, attended a dinner with Petraeus and Karzai after the news conference, officials said.

Gates is due to travel to Germany and Brussels on Wednesday ahead of a meeting of NATO defence ministers from Thursday, where the war in Afghanistan and the current crisis in Libya are expected to top the agenda.

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