Mr. Tillerson’s decision is likely to be welcomed by the Republican majority on Capitol Hill. The chairman of the Senate Foreign Relations Committee, Bob Corker, a Tennessee Republican, said in an interview on Wednesday that he applauded the move to lift the human rights restrictions. He said arms sales should be decided by American strategic needs, and not be mixed with pressure on allies to change their domestic behavior.

“This type of conditionality would be unprecedented and counterproductive to maintaining security cooperation and ultimately addressing human rights issues,” Mr. Corker said. “There are more effective ways to seek changes in partner policies than publicly conditioning weapons transfers in this manner.”

Mr. Tillerson, the former chief executive of Exxon Mobil, has taken on much of the diplomacy with the Gulf Arab states himself, often bypassing American ambassadors and other American officials in the region. A Trump administration official said Mr. Tillerson knew many of the regional players from his time at Exxon Mobil.

The decision on Bahrain also suggests that Mr. Tillerson is likely to deal similarly with Saudi Arabia, the largest and most powerful Sunni force in the region. The Obama administration deepened its rift with its Gulf allies in December over the conflict in Yemen when it blocked a transfer of precision munitions to Saudi Arabia because of concerns about civilian casualties that American officials attributed to poor targeting.

But Mr. Tillerson has signaled he favors reversing that decision, and allowing Raytheon to sell the Saudis about 16,000 guided munitions kits, which upgrade so-called dumb bombs to smart bombs that can more accurately hit targets. The kits, if purchased over the life of the proposed contract, are valued around $350 million. Mr. Tillerson has argued that if civilian casualties are the concern, it makes no sense to deprive the Saudis of precision weaponry.

The new secretary of state was criticized this month for skipping the release of his department’s annual human rights report, an event his Democratic and Republican predecessors used as a moment to pressure allies and adversaries alike by highlighting abuses. During his confirmation hearing, Mr. Tillerson declined to criticize the state-ordered killings in the Philippines or repression in Saudi Arabia, saying he had to make his own assessment of the facts, and could not trust what he read in news reports.

But the sale of F-16s to Bahrain was the first test of whether the Trump administration would reverse the efforts by former President Barack Obama to use America’s main leverage — military support — to force domestic political change in the tiny Gulf state. For weeks, Mr. Tillerson has been talking to members of Congress about easing the restrictions to allow the $2.8 billion sale of fighter jets, and a separate $1 billion deal to support the existing fleet of aircraft.