Suspected “straw” purchasers for the network ended up acquiring about 2,000 guns, most of which are presumed to have reached drug gangs. In December 2010, two weapons that had been bought by one of the suspects were found at the site of a shootout in which a Border Patrol agent, Brian Terry, was killed, setting off the scandal.

Some Republicans and conservative commentators have claimed that senior Obama administration officials must have initiated or authorized the tactics. But public testimony and documents have not produced evidence to support such claims. Information has emerged, however, showing that the Phoenix division of the A.T.F. had a running dispute with Arizona-based prosecutors over how much evidence was necessary to bring charges in straw-purchasing cases, and that its agents had used similar tactics — and lost track of guns — in three other investigations, during the Bush administration.

While the documents in dispute date from 2011, after Fast and Furious was over, Republicans framed their move as being about getting answers and justice for Agent Terry’s family. “It’s our responsibility to investigate when things go wrong, and things went wrong — an agent of the United States was murdered,” said Representative John L. Mica, Republican of Florida, accusing Mr. Holder of “showing contempt for the Congress.”

The Obama administration has been preparing a letter to the House speaker, John A. Boehner, saying that it will not prosecute Mr. Holder for criminal contempt because the Justice Department does not consider it to be a crime to fail to provide information over which a president has asserted executive privilege, officials familiar with the discussions said.

Four more Democrats — and all Republicans — voted for a separate resolution authorizing a lawsuit that would ask a judge to order the Justice Department to comply with its subpoena, setting up a test of Mr. Obama’s assertion of a form of executive privilege that protects agency deliberations from disclosure.

The Justice Department had offered to give Congress several hundred of the disputed documents if Republicans scrapped the contempt recommendation. The White House on Tuesday allowed Republican staff members to scan about a dozen of them, which it portrayed as a representative sample. But the two sides failed to reach a deal.

“My efforts to resolve this matter short of such a battle were rebuffed by Congressman Issa and his supporters,” Mr. Holder said after the vote. “It’s clear that they were not interested in bringing an end to this dispute or obtaining the information they claimed to seek.”