WASHINGTON — Pakistan told the United States that it would reopen NATO’s supply routes into neighboring Afghanistan after Secretary of State Hillary Rodham Clinton said she was sorry for the deaths of two dozen Pakistani soldiers in US airstrikes in November, officials from the two countries said Tuesday.

The agreement ended a bitter seven-month stalemate that threatened to jeopardize counterterrorism cooperation, complicated the US troop withdrawal from Afghanistan and cost the United States more than $1 billion in extra shipping fees as a result of having to use an alternative route through Central Asia.

Clinton said that in a telephone call to Pakistan’s foreign minister, Hina Rabbani Khar, they had agreed that both sides made mistakes that led to the fatal airstrikes.

‘‘We are sorry for the losses suffered by the Pakistani military,’’ Clinton said in a statement that the State Department issued but that officials said had been coordinated with her Pakistani counterpart. ‘‘We are committed to working closely with Pakistan and Afghanistan to prevent this from ever happening again.’’

The accord came together Monday in Islamabad after weeks of behind-the-scenes phone calls, e-mails and meetings between one of Clinton’s deputies, Thomas R. Nides, and a top Pakistani diplomat, US and Pakistani officials said. The agreement reflected a growing realization by Pakistani officials that they had overplayed their hand, and a recognition on both sides that the impasse risked transforming an often rocky relationship into a permanently toxic one at a critical time.

Clinton and her top aides, working closely with senior White House and Pentagon officials, carefully calibrated what she would say in her phone call to Khar to avoid an explicit mention of what one top State Department official called ‘‘the A-word’’ — ‘‘apology.’’ Instead, Clinton opted for the softer ‘‘sorry’’ to meet Pakistan’s longstanding demand for a more formal apology for the airstrikes.

Still, the deal carries risks for both governments. Critics of Pakistan’s weak civilian leadership assailed the accord as a sellout to the United States, and it offers potential fodder for Republicans who contend that President Barack Obama says ‘‘sorry’’ too readily.

‘‘The apology will lower the temperature on US-Pakistan relations,’’ said Shamila N. Chaudhary, a South Asia analyst at the Eurasia Group who served as the director for Pakistan and Afghanistan at the National Security Council. ‘‘However, relations are not on the mend. They remain very much broken and will remain so unless the two countries resolve broader policy differences on Afghanistan.’’

As part of the agreement, Pakistan dropped its insistence on a higher transit fee for each truck carrying NATO’s nonlethal supplies from Pakistan into Afghanistan, after initially demanding as much as $5,000 for each truck.

In the end, Pakistan agreed to keep the fee at the current rate, $250. In return, the White House will ask Congress to reimburse Pakistani about $1.2 billion for costs incurred by 150,000 Pakistani troops carrying out counterinsurgency operations along the border with Afghanistan, a senior US official said.

The November airstrikes, which hit in Pakistani territory in response to reports of militant activity in the area, killed 24 soldiers. In response, Pakistan closed the supply lines and worsened relations already frayed by the shooting death of two Pakistanis by a CIA security contractor and by the Navy SEAL raid that killed Osama bin Laden.

In Pakistan on Tuesday, the decision to reopen the supply routes was met with a general sense of befuddlement and muted criticism that the government had given up a much-trumpeted increase in transit fees for NATO trucks.

But government officials were at pains to say that the accord had never hinged on higher fees. ‘‘I am glad that this breakthrough is not part of any transaction,’’ said Sherry Rehman, Pakistan’s ambassador to the United States. ‘‘We are playing our role as responsible global partner in stabilizing the region.’’

Still, opposition politicians criticized the move and demanded more of an explanation from the Pakistani government and military.