A team of about 20 U.S. troops and aid workers who landed Wednesday on Iraq’s Mount Sinjar determined that a rescue operation of besieged minorities stranded there is probably unnecessary, the Pentagon said.

“There are far fewer” refugees left at the northern Iraq location, where tens of thousands were said to have been surrounded by Sunni Muslim extremists, and they “are in better condition than previously believed,” a Pentagon statement said. It said that humanitarian airdrops and the nightly evacuation of Yazidis on land routes appeared to have lessened the emergency.

However, Kurdish officials and Yazidi refugees said Thursday that thousands of desperate Yazidis remain trapped on the mountain. They said those still stranded on the barren, rocky slopes of Mount Sinjar are mostly the elderly, sick and very young, who were too weak to continue the grueling trek to safety in Iraq’s northern Kurdistan region and were left behind by their relatives.

U.S. officials credited U.S. airstrikes on militant positions surrounding the mountains with allowing thousands to exit. “President said we’re going to break the siege of this mountain, and we broke that siege,” Assistant Secretary of State Brett McGurk said on Twitter.

The Obama administration had been weighing the use of U.S. ground forces and aircraft to mount an emergency rescue of the members of the Yazidi minority sect, even as it seeks to develop a longer-term strategy to push back advances made by Sunni Muslim extremists.

That strategy depends on the formation of a new Iraqi government that is responsive to the concerns of all ethnic and religious groups, an effort that continued its rocky progress Wednesday.

But the strategy also hinges on other variables, including the capabilities of both Iraqi and Kurdish troops, and their capacity to cooperate on the ground. European and other allies are being enlisted to provide weapons and other support, and neighboring Sunni states must be persuaded to use their influence with Iraqi Sunnis who have been reluctant to join the fight against Islamic State militants.

Once a new government is in place, “we will be providing training and equipping, security assistance and advice to Iraqi and Kurdish forces, and then we can begin to squeeze the space where ISIL is operating and start to push them back,” deputy national security adviser Benjamin J. Rhodes said, referring to the group also known as the Islamic State of Iraq and the Levant.

The one “limiting factor,” he said, “is we don’t want to be reintroducing U.S. forces into a combat role on the ground.” Rhodes spoke to reporters in Martha’s Vineyard, Mass., where President Obama is vacationing.

An expansion of U.S. military assistance to counter the Islamic State would almost certainly include the deployment of additional military advisers to Iraq, increased weapons transfers and possibly expanded airstrikes, although Pentagon officials said Wednesday that they have yet to reach a consensus on what such a mix would look like.

The departure of current Prime Minister Nouri al-Maliki will not by itself trigger an influx of military aid or more U.S. trainers, said senior Defense Department officials who were not authorized to discuss internal planning. Instead, the administration is expected to wait until a new government demonstrates clear signs of support from Iraq’s many political and religious factions.

In the meantime, Obama last week authorized the U.S. military to carry out airstrikes on a highly restricted basis — only to prevent the massacre of Iraqi minorities or to neutralize threats to U.S. personnel or property. Defense officials said they do not have clearance, for instance, to target individual Islamic State leaders.

Trapped on Sinjar Mountain

Several Pentagon officials said those restrictions will probably be loosened if Obama approves a broader counterterrorism mission in Iraq. But there is considerable internal debate over the strategic value of a bombing campaign.

Some military officials said an effective air campaign would require the deployment of Special Forces on the ground to help guide attacks on specific targets. Others said technological advances with drones and airborne sensors might preclude the need for spotters on the ground.

Airstrikes over the past several days have targeted militants gathered in the vicinity of Mount Sinjar, where members of the minority Yazidi sect had been stranded. Obama this week deployed an additional 130 U.S. troops to Irbil, the capital of Iraq’s northern Kurdish region, to assess the situation and plan for an expanded humanitarian relief mission.

On Wednesday, a team of those troops, accompanied by Agency for International Development officials from Irbil, flew to Mount Sinjar to inspect conditions — the first time U.S. forces have been present on the ground there, according to a Defense Department official. The team safely returned to Irbil later in the day.

The new military deployments brought to more than 900 the total number of U.S. troops Obama has authorized to protect U.S. personnel and facilities in Irbil and Baghdad and to assess the needs and capabilities of Iraqi forces. Of those, 864 are on the ground in Iraq, and none of them has fired a weapon or been involved in a firefight, Pentagon spokesman Col. Steve Warren said.

The new troops in the Kurdish region are accompanied by four V-22 Osprey tilt-rotor aircraft and an unspecified number of helicopters, the first time the Pentagon has acknowledged stationing aircraft in northern Iraq. The Pentagon did not say what aircraft had carried the team to Mount Sinjar.

Before the assessment, Rhodes, the White House adviser, had said that a number of options were under consideration for a rescue, including airlifts and establishing a safe corridor on the ground.

“You look at corridors, you look at airlifts, you look at different ways to move people who are in a very dangerous place on that mountain to a safer position,” Rhodes said. The Pentagon later said that “the Yazidis who remain” on the mountain “are in better condition than previously believed and continue to have access to the food and water that we have dropped.” The statement said humanitarian assistance would continue as the remainder leave the area.

Late Wednesday, after officials decided against organizing an evacuation, U.S. military airplanes dropped more water and food supplies on the mountain — the seventh airdrop in a week, the U.S. Central Command announced.

Pressed in separate briefings on whether any of the U.S. troops in Iraq could be involved in combat, both Rhodes and Warren said the troops were equipped to defend themselves but were not there to launch offensive actions.

The effectiveness of Iraq’s security forces has been eroded badly since the U.S. military withdrew from the country in December 2011. Pentagon officials said many Iraqi units needed basic retraining in combat skills and in planning offensive operations before they could be expected to take on Islamic State fighters, who in many cases are better equipped and more experienced.

The Kurdish region’s separate military force, known as the pesh merga, lacks heavy weaponry and supplies. The two forces have long been at odds, as the regional government in Irbil has argued over autonomy and fair distribution of national resources with the central government in Baghdad under Maliki.

Although some covert assistance from the United States is said to have arrived for the pesh merga, the official position of the United States and other potential suppliers is that all aid must first go through Baghdad.

That limitation appears to have been eased in recent days — the Iraqi government has sent some of its own supplies to Irbil and has authorized outsiders to send weapons and other supplies directly to the Kurds.

Europe, which is increasingly worried about its own passport-holders among Islamic State fighters, has dramatically stepped up its contribution to the conflict this week.

France said Wednesday that it would start supplying unspecified arms directly to the Kurdish forces, in coordination with Baghdad. Britain, which has dropped humanitarian supplies to the Yazidis, said it would transport military equipment supplied by others to the Kurds.

German Foreign Minister Frank-Walter Steinmeier, in a Wednesday interview with ZDF radio, said that “if the current threat level persists, I cannot rule out that we will have to deliver weapons.”

Islamic State forces have advanced through western Iraq, southward to within 60 miles of Baghdad, in part because Sunni Arab populations in those areas, opposed to Maliki’s Shiite-led government, have tolerated or even assisted them.

As the militants’ offensive began in June, Arab states in the region with influence over the Sunnis — including Saudi Arabia, the United Arab Emirates, Kuwait and Jordan — strongly advised the Obama administration not to intervene on the Iraqi government’s behalf lest the United States be viewed as supporting Maliki, according to Arab officials who spoke on the condition of anonymity to describe those conversations.

As the administration began to publicly pivot away from Maliki in recent weeks, those governments offered assurances that they would take action. Once Maliki is gone and a more representative, inclusive government is established, officials said, Sunni tribes in western Iraq will be prepared to take on the Islamic State.

But “it’s not a magic wand where you can deal with this issue in days or a week,” an Arab official said. “It will take a long time.”

Liz Sly in Irbil and Katie Zezima in Martha’s Vineyard contributed to this report.