After Congress failed to approve more visas in the House and Senate defense bills, a bipartisan group of senators tried again last month with a compromise that would have added 2,500 visas — only to be blocked by Senator Mike Lee, Republican of Utah, who said his quarrel was not with the visa program but rather with the fact that it was getting a vote while one of his own, unrelated measures was not.

Mr. McCain responded by accusing Mr. Lee of “signing the death warrants” of people who had put their lives on the line to help the United States.

Mr. Stanikzai would agree. Now 24, he first applied in 2013 for one of the visas, writing to American officials later that year that an imam had warned his father that his son must stop working for the Americans or be killed. He then described how he came under fire as he drove home from his mosque, his car hit by three rounds from an AK-47, he said.

If the Taliban find him — or any of the Afghans hoping the United States will grant them visas — “they will kill us,” he said in an interview.

Yet with neither the House nor the Senate approving more visas in their competing defense policy bills, lawmakers hashing out a compromise this fall may be unlikely to step in.

Much of the resistance seems to stem from a growing discomfort with immigrants, said Senator Jeanne Shaheen, Democrat of New Hampshire and one of the visa program’s most active supporters. One Republican counterproposal this year would have offset each special visa by deducting one green-card visa from the 50,000 allocated every year to ordinary immigrants. “What that said to me is that the issue here is not really the special immigrant visas for our interpreters,” she said. “It was, ‘How do we keep from allowing more people into the United States?’ ”