The goal of the group, according to internal documents cited by the F.B.I., was to persuade the United States government that it was in its interest to push India to allow a vote in Kashmir to decide its future. The group’s strategy was to offset the Indian lobby by targeting members of the Congressional committees that focus on foreign affairs with private briefings and events, staging activities that would draw media attention and otherwise to elevate the issue of Kashmir — the disputed region between India and Pakistan that each country controls in part but claims entirely — in Washington.

The F.B.I. said that there was no evidence that any of the lawmakers who received campaign funds from Pakistan were aware of its origins, and it did not name any of the recipients.

However, a search in Federal Elections Commission databases for contributions by Mr. Fai showed that he has made more than $20,000 in campaign contributions over the past two decades. The bulk of his donations went to two recipients: the National Republican Senatorial Committee and Representative Dan Burton, a Republican from Indiana.

Mr. Fai made numerous — though smaller — contributions to Democrats as well, including to Representatives James P. Moran of Virginia, Dennis J. Kucinich of Ohio and Gregory W. Meeks of New York, and $250 donations to the 2000 and 2008 presidential campaigns of Al Gore and Barack Obama.

Mr. Ahmad also donated to Mr. Burton, records show. For at least 15 years, Mr. Burton has been a champion for Kashmiri causes in Congress, appealing to Presidents Bill Clinton and Obama to get more involved in attempting to mediate a settlement between India and Pakistan over the border region. He has also endorsed allowing the Kashmiri people to vote on their own fate.

Image Representative Dan Burton of Indiana is another politician who may have unknowingly received money funneled from Pakistan's military. Credit... Charles Dharapak/Associated Press

Mr. Burton said he was “deeply shocked” by the arrest of Mr. Fai, because he had known him for 20 years and “in that time I had no inkling of his involvement with any foreign intelligence operation and had presumed our correspondence was legitimate.” He said he would donate the funds provided to his campaign to the Boy Scouts of America.