WASHINGTON — When Pakistani forces freed a Canadian-American family this fall held captive by militants, they also captured one of the abductors. United States officials saw a potential windfall: He was a member of the Taliban-linked Haqqani network who could perhaps provide valuable information about at least one other American hostage.

The Americans demanded access to the man, but Pakistani officials rejected those requests, the latest disagreement in the increasingly dysfunctional relationship between the countries. Now, the Trump administration is strongly considering whether to withhold $255 million in aid that it had delayed sending to Islamabad, according to American officials, as a show of dissatisfaction with Pakistan’s broader intransigence toward confronting the terrorist networks that operate there.

The administration’s internal debate over whether to deny Pakistan the money is a test of whether President Trump will deliver on his threat to punish Islamabad for failing to cooperate on counterterrorism operations. Relations between the United States and Pakistan, long vital for both, have chilled steadily since the president declared over the summer that Pakistan “gives safe haven to agents of chaos, violence and terror.”

The United States, which has provided Pakistan more than $33 billion in aid since 2002, said in August that it was withholding the $255 million until Pakistan did more to crack down on internal terrorist groups. Senior administration officials met this month to decide what to do about the money, and American officials said a final decision could be made in the coming weeks.