WASHINGTON — The American ambassador to Algeria and senior counterterrorism officials have proposed sharing more information with Algerian security forces to help them kill or capture militants in their own country and in areas just across their borders.

Their approach reflects the growing support within the administration for more forceful action against extremists in the area since the attack on a gas field in eastern Algeria last month left 37 dead, including three Americans, and focused new concerns on terrorist activity in Africa.

Under one plan, information from American surveillance drones would be provided to Algerian forces to enable them to engage in operations both inside Algeria and possibly, in a limited way, across its borders. The United States is already providing surveillance information to the French-led military operation in Mali to help combat militants there who last year seized the northern half of the country.

In a cable to the State Department last week, according to administration officials, Henry S. Ensher, the United States envoy in Algiers, urged that the pursuit of the Algerian militant Mokhtar Belmokhtar, the mastermind of the gas field attack, be made a priority. Toward that end, he recommended that the Obama administration tell the Algerians that if they allowed the United States to fly unarmed drones over the border area of Algeria as well as over Mali, the Americans would share the information with the Algerian government.