Along party lines, the Senate voted 52 to 45 on a key procedural motion, failing to reach the 60 votes needed to begin debating the measure. Senator Olympia J. Snowe, the moderate Republican from Maine who is retiring, voted present. Senators said quiet negotiations had begun to resolve the impasse, but Democrats sought to raise the political pressure, vowing to take to the Senate floor to show the cost of inaction for students in their states. “Mitt Romney says he supports what we’re trying to do. I’d suggest he pick up the phone and call Senator McConnell,” said Senator Harry Reid of Nevada, the Senate majority leader, referring to the Senate Republican leader, Mitch McConnell of Kentucky. Republicans blamed Democrats for the impasse and suggested that they were manufacturing a political controversy instead of working out differences in private. “We all agree we’re not going to let the rate go up,” Mr. McConnell said. The vote was the Senate Republicans’ 21st successful filibuster of a Democratic bill this Congress, which started in January 2011. Republicans have blocked consideration of President Obama’s full jobs proposal, as well as legislation repealing tax breaks for oil companies, helping local governments pay teachers and first responders, and setting a minimum tax rate for households earning more than $1 million a year. Republicans say the measures were flawed and potentially harmful to the economic recovery. But the student loan filibuster may be the highest-profile stalemate yet, because unlike those earlier bills, this one is not likely to be abandoned. Mr. Obama has elevated the issue by hammering Republicans on it for weeks. American students took out twice the value of student loans in 2011, about $112 billion, as they did a decade before, after adjusting for inflation. Over all, Americans now owe about $1 trillion in student loans. In 2010, such debt surpassed credit card debt for the first time. The bill in limbo addresses only part of that burden. Graduate students with Stafford loans pay a higher rate, as do students with unsubsidized Stafford loans. Most undergraduates take out both unsubsidized and subsidized loans. Republicans say they want to extend Democratic legislation passed in 2007 that temporarily reduced interest rates for low- and middle-income undergraduates who receive subsidized Stafford loans to 3.4 percent from 6.8 percent. But the Republicans would not accept the Senate Democrats’ proposal to pay for a one-year extension by changing a law that allows some wealthy taxpayers to avoid paying Social Security and Medicare taxes by classifying their pay as dividends, not cash income. “They want to raise taxes on people who are creating jobs when we are still recovering from the greatest recession since the Great Depression,” said Senator Lamar Alexander, Republican of Tennessee, who instead wanted to pay for the rate decrease by eliminating a fund for preventive health care in Mr. Obama’s health care law. Before the vote, Senate Democrats arrayed college students to plead for a yes vote, including Clarise McCants, 21, a junior at Howard University in Washington who said she pulled herself out of a troubled neighborhood in North Philadelphia and relies on $13,500 in Stafford loans for her tuition. “I know I’m not the only one with dreams,” she said. “I’m here to ask Congress, ‘Don’t double my rate.’ ” Republicans have not always been so averse to closing the loophole that the Senate bill addresses. In 2004, when it emerged that John Edwards, then a vice-presidential hopeful, had classified himself as a “subchapter S corporation” to pay himself dividends rather than income, conservatives criticized him for avoiding payroll taxes. But the Democratic line of attack has been complicated by the House’s actions. Shrugging off a veto threat, the House passed an extension of the subsidized rate last month, paid for with the preventive health care fund. Thirteen Democrats voted for the bill, making up for the 30 Republicans who voted no because they opposed federal subsidies for an interest rate that they believed should be set by market forces. Those Democratic defections put the House bill over the top and fortified Republican arguments that the Senate Democrats were now to blame for the stalemate. Representative Steny Hoyer of Maryland, the House minority whip, said Tuesday that those Democratic votes were driven by politics, not substance. “They didn’t want that 30-second ad” attacking them for opposing a rate-subsidy extension, he said. “That was not a demonstration at all for the funding source.” Republicans made clear they would go on offense, blaming Democrats if interest rates doubled July 1. “Instead of compounding the problem with more bad policies that raise taxes on small businesses and raid Social Security and Medicare, we must work together to prevent a rate increase on students and make it easier for job creators to hire them when they graduate,” Senator Roy Blunt, Republican of Missouri, said after the vote.