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But the surprise French assault last Friday to blunt the Islamists' advance upended those plans and set off a cascading series of events, culminating in a raid on Wednesday by militants on a foreign-run gas field in Algeria. That attack threatens to widen the violence in an impoverished region and drag Western governments deeper into combating an incipient insurgency.

And yet the rush of events has masked the fact that officials in Washington still have only an impressionistic understanding of the militant groups that have established a safe haven in Mali, and they are deeply divided about whether some of these groups even pose a threat to the United States.

Moreover, the hostage situation in Algeria has only heightened concerns that a Western military intervention could transform militant groups that once had only a regional focus into avowed enemies of the United States — in other words, that the backlash might end up being worse than the original threat.

Largely for these reasons, the Obama administration adopted a strategy over the past year to contain the Islamists in Mali until African troops were ready to confront them, rather than to challenge them directly with an American military campaign of drone strikes or commando raids.

During Congressional testimony in June, Johnnie Carson, the assistant secretary of state for African affairs, played down the terrorist threat to the United States from Mali, saying that the Qaeda affiliate operating there "has not demonstrated the capability to threaten U.S. interests outside of West or North Africa, and it has not threatened to attack the U.S. homeland."

Some Pentagon officials have long taken a more hawkish stance, and they cite intelligence reports that fighters with ties to Al Qaeda in the Islamic Maghreb, which has a loose affiliation to the remnants of Osama bin Laden's terrorist network, played a role in the deadly attack in September on the American diplomatic compound in Benghazi, Libya. They have pushed for targeted strikes against Islamist leaders in northern Mali, arguing that killing the leadership could permanently cripple the strength of the militants.

The administration has embraced a targeted killing strategy elsewhere, notably in Pakistan, Yemen and Somalia, after top White House, Pentagon and C.I.A. officials determined that militants in those countries were bent on attacking the United States.

Asked if fighters from Al Qaeda in the Islamic Maghreb posed such an imminent threat, Gen. Carter F. Ham, the top American commander in Africa, said, "Probably not." But, he said in an interview, "they subscribe to Al Qaeda's ideology" and have said that their intent is to attack Westerners in Europe and, "if they could, back to the United States."

Defense Secretary Leon E. Panetta made it clear on Wednesday that he considered the group a serious danger. "This is an Al Qaeda operation," he told reporters while traveling in Italy, "and it is for that reason that we have always been concerned about their presence in Mali, because they would use it as a base of operations to do exactly what happened in Algeria."

It is too early to judge the impact of the French-led offensive in Mali, which came after an urgent plea by Mali's government for help in repelling Islamist fighters who were rapidly moving south. But on Wednesday, some American officials said that the hostage episode in Algeria could be just the beginning of a wave of attacks against foreigners in the region. And there is a chance that if the Americans taken hostage on Wednesday are killed by their captors, American officials might reconsider their pledges not to commit ground troops to the battle.