NEW DELHI — The United States lost a hard-fought competition to supply a new generation of fighter jets to India, which has listed two European manufacturers as the finalists for an order estimated to be worth $10 billion.

The decision was a blow for President Obama, who had pushed hard for this and other defense deals during his visit to India in November as part of his agenda to deepen and broaden the United States’ relationship with India. The American ambassador to India, Timothy J. Roemer, who separately announced on Thursday that he would resign from his post for personal reasons, said the United States was “deeply disappointed by this news.”

While political and economic relations between India and the United States have been warming for years, American arms makers have struggled to win big contracts here. After decades of frosty relations during the cold war, which pushed India to rely extensively on the Soviet Union for military hardware, many in the Indian defense establishment are still wary of American intentions and United States military aid to Pakistan, India’s main adversary.

The American bid to build the fighters came from Boeing and Lockheed Martin. Boeing had offered its F/A -18 jets, and Lockheed Martin pitched its F-16 planes. But India instead narrowed the list to the Rafale fighter from Dassault and the Eurofighter Typhoon jet made by a consortium of European companies. Russian and Swedish bids were also turned down.