WASHINGTON — White House-backed legislation to renew jobless benefits unexpectedly cleared an initial Senate hurdle today, raising the prospect of a mid-winter compromise to ease the impact of the recession on the long-term unemployed.

“Let’s get it done,” President Barack Obama exhorted lawmakers at the White House shortly after the vote.

The vote was 60-37 to limit debate on the three-month legislation, with a half-dozen Republicans siding with the Democrats on the test vote.

At the same time, the Republican leader, Sen. Mitch McConnell of Kentucky, said he and his rank and file would seek changes so the bill’s $6.4 billion cost would not add to deficits.

Senate Democrats have so far rejected that approach, although there were signs they would eventually yield.

Shortly after the Senate vote, House Speaker John Boehner, R-Ohio, issued a statement expressing views similar to McConnell’s. Almost simultaneously, a senior Senate Democrat, Chuck Schumer of New York, signaled a willingness to consider changes to offset the impact of the bill on the deficit, calling that “the second best option.”